Distances in Graph Theory

Abstract

A graph is a pair G = (V, E), where V is a set, called the set of vertices of the graph G, and E is a set of unordered pairs of vertices, called the edges of the graph G. A directed graph (or digraph) is a pair D = (V, E), where V is a set, called the set of vertices of the digraph D, and E is a set of ordered pairs of vertices, called arcs of the digraph D.

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A graph is a pair G = (V, E), where V is a set, called the set of vertices of the graph G, and E is a set of unordered pairs of vertices, called the edges of the graph G. A directed graph (or digraph) is a pair D = (V, E), where V is a set, called the set of vertices of the digraph D, and E is a set of ordered pairs of vertices, called arcs of the digraph D.

A graph in which at most one edge may connect any two vertices, is called a simple graph. If multiple edges are allowed between vertices, the graph is called a multigraph. A graph, together with a function which assigns a positive weight to each edge, is called a weighted graph or network.

The graph is called finite (infinite) if the set V of its vertices is finite (infinite, respectively). The order and size of a finite graph (V, E) are | V | and | E | , respectively.

A subgraph of a graph G = (V, E) is a graph $G^{^{{\prime}} } = (V ^{^{{\prime}} },E^{^{{\prime}} })$ with $V ^{^{{\prime}} } \subset V$ and $E^{^{{\prime}} } \subset E$. If $G^{^{{\prime}} }$ is a subgraph of G, then G is called a supergraph of $G^{^{{\prime}} }$. A subgraph $(V ^{^{{\prime}} },E^{^{{\prime}} })$ of (V, E) is its induced subgraph if $E^{^{{\prime}} } =\{ e = \mathit{uv} \in E: u,v \in V ^{^{{\prime}} }\}$.

A graph G = (V, E) is called connected if, for any u, v ∈ V, there exists a (u − v) walk, i.e., a sequence of edges uw ₁ = w ₀ w ₁, w ₁ w ₂, $\ldots$, w _n−1 w _n  = w _n−1 v from E. A (u − v) path is a (u − v) walk with distinct edges. A graph is called m-connected if there is no set of m − 1 edges whose removal disconnects the graph; a connected graph is 1-connected. A digraph D = (V, E) is called strongly connected if, for any u, v ∈ V, the directed (u − v) and (v − u) paths both exist. A maximal connected subgraph of a graph G is called its connected component.

Vertices connected by an edge are called adjacent. The degree d e g(v) of a vertex v ∈ V of a graph G = (V, E) is the number of its vertices adjacent to v.

A complete graph is a graph in which each pair of vertices is connected by an edge. A bipartite graph is a graph in which the set V of vertices is decomposed into two disjoint subsets so that no two vertices within the same subset are adjacent. A simple path is a simple connected graph in which two vertices have degree 1, and other vertices (if they exist) have degree 2; the length of a path is the number of its edges.

A cycle is a closed simple path, i.e., a simple connected graph in which every vertex has degree 2. The circumference of a graph is the length of the longest cycle in it. A tree is a simple connected graph without cycles. A tree having a path from which every vertex has distance ≤ 1 or ≤ 2, is called a caterpillar or lobster, respectively.

Two graphs which contain the same number of vertices connected in the same way are called isomorphic. Formally, two graphs G = (V (G), E(G)) and H = (V (H), E(H)) are called isomorphic if there is a bijection f: V (G) → V (H) such that, for any u, v ∈ V (G), uv ∈ E(G) if and only if f(u)f(v) ∈ E(H).

We will consider mainly simple finite graphs and digraphs; more exactly, the equivalence classes of such isomorphic graphs.

1 Distances on the Vertices of a Graph

• Path metric

  The path metric (or graphic metric , shortest path metric) d _path is a metric on the vertex-set V of a connected graph G = (V, E) defined, for any u, v ∈ V, as the length of a shortest (u − v) path in G, i.e., a geodesic. Examples follow.

  Given an integer n ≥ 1, the line metric on $\{1,\ldots,n\}$ in Chap. 1 is the path metric of the path $P_{n} =\{ 1,\ldots,n\}$. The path metric of the Cayley graph $\Gamma$ of a finitely generated group (G, ⋅ , e) is called a word metric.

  The hypercube metric is the path metric of a hypercube graph H(m, 2) with the vertex-set V = { 0, 1}^m, and whose edges are the pairs of vectors x, y ∈ { 0, 1}^m such that $\vert \{i \in \{ 1,\ldots,n\}: x_{i}\neq y_{i}\}\vert = 1$; it is equal to $\vert \{i \in \{ 1,\ldots,n\}: x_{i} = 1\}\bigtriangleup \{i \in \{ 1,\ldots,n\}: y_{i} = 1\}\vert$. The graphic metric space associated with a hypercube graph coincides with a Hamming cube, i.e., the metric space $(\{0,1\}^{m},d_{l_{1}})$.

  The belt distance (Garber–Dolbilin, 2010) is the path metric of a belt graph B(P) of a polytope P with centrally symmetric facets. The vertices of B(P) are the facets of P and two vertices are connected by an edge if the corresponding facets lie in the same belt (the set of all facets of P parallel to a given face of codimension 2).

  The reciprocal path metric is called geodesic similarity .

• Weighted path metric

  The weighted path metric d _wpath is a metric on the vertex-set V of a connected weighted graph G = (V, E) with positive edge-weights (w(e)) _e ∈ E defined by

  $$\displaystyle{\min _{P}\sum _{e\in P}w(e),}$$

  where the minimum is taken over all (u − v) paths P in G.

  Sometimes, $\frac{1} {w(e)}$ is called the length of the edge e. In the theory of electrical networks, the edge-length $\frac{1} {w(e)}$ is identified with the resistance of the edge e. The inverse weighted path metric is $\min _{P}\sum _{e\in P} \frac{1} {w(e)}$.

• Metric graph

  A metric (or metrized) graph is a connected graph G = (V, E), where edges e are identified with line segments [0, l(e)] of length l(e). Let x _e be the coordinate on the segment [0, l(e)] with vertices corresponding to x _e  = 0, l(e); the ends of distinct segments are identified if they correspond to the same vertex of G. A function f on G is the | E | -tuple of functions f _e (x _e ) on the segments.

  A metric graph can be seen as an infinite metric space (X, d), where X is the set of all points on above segments, and the distance between two points is the length of the shortest, along the line segments traversed, path connecting them. Also, it can be seen as one-dimensional Riemannian manifold with singularities.

  There is a bijection between the metric graphs, the equivalence classes of finite connected edge-weighted graphs and the resistive electrical networks: if an edge e of a metric graph has length l(e), then $\frac{1} {l(e)}$ is the weight of e in the corresponding edge-weighted graph and l(e) is the resistance along e in the corresponding resistive electric circuit. Cf. the resistance metric.

  A quantum graph is a metric graph equipped with a self-adjoint differential operator (such as a Laplacian) acting on functions on the graph. The Hilbert space of the graph is ⊕ _e ∈ E L ²([0, w(e)]), where the inner product of functions is $\langle f,g\rangle =\sum _{e\in E}\int _{0}^{w(e)}f_{e}^{{\ast}}(x_{e})g_{e}(x_{e})\mathit{dx}_{e}$.

• Spin network

  A spin network is (Penrose, 1971) a connected graph (V, E) with edge-weights (w(e)) _e ∈ E (spins), $w(e) \in \mathbb{N}$, such that for any distinct edges e ₁, e ₂, e ₃ with a common vertex, it holds spin triangle inequality $\vert w(e_{1}) - w(e_{2})\vert \leq w(e_{3}) \leq w(e_{1}) + w(e_{2})$ and fermion conservation: w(e ₁) + w(e ₂) + w(e ₃) is an even number.

  The quantum space-time (Chap. 24) in Loop Quantum Gravity is a network of loops at Planck scale. Loops are represented by adapted spin networks: directed graphs whose arcs are labeled by irreducible representations of a compact Lie group and vertices are labeled by interwinning operators from the tensor product of labels on incoming arcs to the tensor product of labels on outgoing arcs. Such networks represent “quantum states” of the gravitational field on a 3D hypersurface.

• Detour distance

  Given a connected graph G = (V, E), the detour distance is (Chartrand and Zhang, 2004) a metric on the vertex-set V defined, for u ≠ v, as the length of the longest (u − v) path in G. So, this distance is 1 or | V | − 1 if and only if uv is a bridge of G or, respectively, G contains a Hamiltonian (u − v) path.

  The monophonic distance is (Santhakumaran and Titus, 2011) a distance (in general, not a metric) on the V defined, for u ≠ v, as the length of a longest monophonic (or minimal), i.e., containing no chords, (u − v) path in G.

  The height of a DAG (acyclic digraph) is the number of vertices in a longest directed path.

• Cutpoint additive metric

  Given a graph G = (V, E), Klein–Zhu, 1998, call a metric d on V graph-geodetic metric if, for u, w, v ∈ V, the triangle equality d(u, w) + d(w, v) = d(u, v) holds if w is a (u,v)-gatekeeper, i.e., w lies on any path connecting u and v. Cf. metric interval in Chap. 1. Any gatekeeper is a cutpoint, i.e., removing it disconnects G and a pivotal point, i.e., it lies on any shortest path between u and v.

  Chebotarev, 2010, call a metric d on the vertices of a multigraph without loops cutpoint additive if d(u, w) + d(w, v) = d(u, v) holds if and only if w lies on any path connecting u and v. The resistance metric is cutpoint additive (Gvishiani and Gurvich, 1992), while the path metric is graph-geodetic only (in the weaker Klein–Zhu sense). See also Chebotarev–Shamis metric.

• Graph boundary

  Given a connected graph G = (V, E), a vertex v ∈ V is (Chartrand et al., 2003) a boundary vertex if there exists a witness, i.e., a vertex u ∈ V such that d(u, v) ≥ d(u, w) for all neighbors w of v. So, the end-vertices of a longest path are boundary vertices. The boundary of G is the set of all boundary vertices.

  The boundary of a subset M ⊂ V is the set ∂ M ⊂ E of edges having precisely one endpoint in M. The isoperimetric number of G is (Buser, 1978) $\inf \frac{\partial M} {\vert M\vert }$, where the infimum is taken over all M ⊂ V with 2 | M | ≤ | V | .

• Graph diameter

  Given a connected graph G = (V, E), its graph diameter is the largest value of the path metric between vertices of G.

  A connected graph is vertex-critical (edge-critical) if deleting any vertex (edge) increases its diameter. A graph G of diameter k is goal-minimal if for every edge uv, the inequality d _G−uv (x, y) > k holds if and only if {u, v} = { x, y}.

  If G is m-connected and a is an integer, 0 ≤ a < m, then the a -fault diameter of G is the maximal diameter of a subgraph of G induced by | V | − a of its vertices. For 0 < a ≤ m, the a -wide distance d _a (u, v) between vertices u and v is the minimum integer l, for which there are at least a internally disjoint (u − v) paths of length at most l in G: cf. Hsu–Lyuu–Flandrin–Li distance. The a-wide diameter of G is max _{u, v ∈ V} d _a (u, v); it is at least the (a − 1)-fault diameter of G.

  Given a strong orientation O of a connected graph G = (V, E), i.e., a strongly connected digraph D = (V, E ^′) with arcs e ^′ ∈ E ^′ obtained from edges e ∈ E by orientation O, the diameter of D is the maximal length of shortest directed (u − v) path in it. The oriented diameter of a graph G is the smallest diameter among strong orientations of G. If it is equal to the diameter of G, then any orientation realizing this equality is called tight. For example, a hypercube graph H(m, 2) admits a tight orientation if m ≥ 4 (McCanna, 1988).

• Path quasi-metric in digraphs

  The path quasi-metric in digraphs d _dpath is a quasi-metric on the vertex-set V of a strongly connected digraph D = (V, E) defined, for any u, v ∈ V, as the length of a shortest directed (u − v) path in D.

  The circular metric in digraphs is a metric on the vertex-set V of a strongly connected digraph D = (V, E), defined by d _dpath (u, v) + d _dpath (v, u).

• Strong distance in digraphs

  The strong distance in digraphs is a metric between vertices v and v of a strongly connected digraph D = (V, E) defined (Chartrand–Erwin–Raines–Zhang, 1999) as the minimum size (the number of edges) of a strongly connected subdigraph of D containing v and v. Cf. Steiner distance of a set.

• $\Upsilon$ -metric

  Given a class $\Upsilon$ of connected graphs, the metric d of a metric space (X, d) is called a $\Upsilon$ -metric if (X, d) is isometric to a subspace of a metric space (V, d _wpath), where $G = (V,E) \in \Upsilon$, and d _wpath is the weighted path metric on V with positive edge-weight function w; cf. tree-like metric.

• Tree-like metric

  A tree-like metric (or weighted tree metric ) d on a set X is a $\Upsilon$ -metric for the class $\Upsilon$ of all trees, i.e., the metric space (X, d) is isometric to a subspace of a metric space (V, d _wpath), where T = (V, E) is a tree, and d _wpath is the weighted path metric on the vertex-set V of T with a positive weight function w. A metric is a tree-like metric if and only if it satisfies the four-point inequality.

  A metric d on a set X is called a relaxed tree-like metric if the set X can be embedded in some (not necessary positively) edge-weighted tree such that, for any x, y ∈ X, d(x, y) is equal to the sum of all edge weights along the (unique) path between corresponding vertices x and y in the tree. A metric is a relaxed tree-like metric if and only if it is a relaxed four-point inequality metric.

• Katz similarity

  Given a connected graph G = (V, E) with positive edge-weight function w = (w(e)) _e ∈ E , let $V =\{ v_{1},\ldots,v_{n}\}$. Denote by A the (n × n)-matrix ((a _ij )), where a _ij  = a _ji  = w(ij) if ij is an edge, and a _ij  = 0, otherwise. Let I be the identity (n × n)-matrix, and let $t,0 < t < \frac{1} {\lambda }$, be a parameter, where λ = max _i  | λ _i  | is the spectral radius of A and λ _i are the eigenvalues of A. Define the (n × n)-matrix

  $$\displaystyle{K = ((k_{\mathit{ij}})) =\sum _{ i=1}^{\infty }t^{i}A^{i} = (I - tA)^{-1} - I.}$$

  The number k _ij is called the Katz similarity between v _i and v _j . Katz, 1953, proposed it for evaluating social status.

  Chebotarev, 2011, defined, for a similar (n × n)-matrix $((c_{\mathit{ij}})) =\sum _{ i=0}^{\infty }t^{i}A^{i} = (I - tA)^{-1}$ and connected edge-weighted multigraphs allowing loops, the walk distance between vertices v _i and v _j as any positive multiple of $d_{t}(i,j) = -\ln \frac{c_{\mathit{ij}}} {\sqrt{c_{\mathit{ii } } \,c_{\mathit{jj }}}}$ (cf. the Nei standard genetic distance in Chap. 23). He proved that d _t is a cutpoint additive metric and the path metric in G coincides with the short walk distance $\lim _{t\rightarrow 0^{+}} \frac{d_{t}} {-\ln t}$ in G, while the resistance metric in G coincides with the long walk distance $\lim _{t\rightarrow \frac{1} {\lambda } ^{-}} \frac{2d_{t}} {n(t^{-1}-\lambda )}$ in the graph G ^′ obtained from G by attaching weighted loops that provide G ^′ with uniform weighted degrees.

  If G is a simple unweighted graph, then A is its adjacency matrix. Let J be the (n × n)-matrix of all ones and let μ = min _i λ _i . Let N = ((n _ij )) = μ(I − J) − A. Neumaier, 1980, remarked that $((\sqrt{n_{\mathit{ij }}}))$ is a semimetric on the vertices of G.

• Resistance metric

  Given a connected graph G = (V, E) with positive edge-weight function w = (w(e)) _e ∈ E , let us interpret the edge-weights as electrical conductances and their inverses as resistances. For any two different vertices u and v, suppose that a battery is connected across them, so that one unit of a current flows in at u and out in v. The voltage (potential) difference, required for this, is, by Ohm’s law, the effective resistance between u and v in an electrical network; it is called the resistance (or electric) metric $\Omega (u,v)$ between them (Sharpe, 1967, Gvishiani–Gurvich, 1987, and Klein–Randic, 1993 [KlRa93]). So, if a potential of one volt is applied across vertices u and v, a current of $\frac{1} {\Omega (u,v)}$ will flow. The number $\frac{1} {\Omega (u,v)}$ is a measure of the connectivity between u and v.

  Let $r(u,v) = \frac{1} {w(e)}$ if uv is an edge, and r(u, v) = 0, otherwise. Formally,

  $$\displaystyle{\Omega (u,v) = (\sum _{w\in V }f(w)r(w,v))^{-1},}$$

  where f: V → [0, 1] is the unique function with f(u) = 1, f(v) = 0 and $\sum _{z\in V }(f(w) - f(z))r(w,z) = 0$ for any w ≠ u, v.

  The resistance metric is a weighted average of the lengths of all (u − v) paths. It is applied when the number of (u − v) paths, for any u, v ∈ V, matters.

  A probabilistic interpretation (Gobel–Jagers, 1974) is: $\Omega (u,v) = (\mathit{deg}(u)\mathit{Pr}(u \rightarrow v))^{-1}$, where deg(u) is the degree of the vertex u, and Pr(u → v) is the probability for a random walk leaving u to arrive at v before returning to u. The expected commuting time between u and v is $2\sum _{e\in E}w(e)\Omega (u,v)$.

  Then $\Omega (u,v) \leq \min _{P}\sum _{e\in P} \frac{1} {w(e)}$, where P is any (u − v) path (cf. inverse weighted path metric), with equality if and only if such a path P is unique. So, if w(e) = 1 for all edges, the equality means that G is a geodetic graph, and hence the path and resistance metrics coincide. Also, it holds that $\Omega (u,v) = \frac{\vert \{t:\mathit{uv}\in t\in T\}\vert } {\vert T\vert }$ if uv is an edge, and $\Omega (u,v) = \frac{\vert T^{{\prime}}-T\vert } {\vert T\vert }$, otherwise, where T, T ^′ are the sets of spanning trees for G = (V, E) and G ^′ = (V, E ∪{uv}).

  If w(e) = 1 for all edges, then $\Omega (u,v) = (g_{\mathit{uu}} + g_{\mathit{vv}}) - (g_{\mathit{uv}} + g_{vu})$, where ((g _ij )) is the Moore–Penrose generalized inverse of the Laplacian matrix ((l _ij )) of the graph G: here l _ii is the degree of vertex i, while, for i ≠ j, l _ij  = 1 if the vertices i and j are adjacent, and l _ij  = 0, otherwise. A symmetric (for an undirected graph) and positive-semidefinite matrix ((g _ij )) admits a representation KK ^T. So, $\Omega (u,v)$ is the squared Euclidean distance between the u-th and v-th rows of K.

  The distance $\sqrt{\Omega (u, v)}$ is a Mahalanobis distance (cf. Chap. 17) with a weighting matrix ((g _ij )). So, $\Omega _{u,v} = a_{\mathit{uv}}\vert ((g_{\mathit{ij}}))\vert a_{\mathit{uv}}$, where a _uv are the vectors of zeros except for + 1 and − 1 in the u-th and v-th positions. This distance is called a diffusion metric in [CLMNWZ05] because it depends on a random walk.

  The number $\frac{1} {2}\sum _{u,v\in V }\Omega (u,v)$ is called the total resistance (or Kirchhoff index) of G.

• Hitting time quasi-metric

  Let G = (V, E) be a connected graph. Consider random walks on G, where at each step the walk moves to a vertex randomly with uniform probability from the neighbors of the current vertex. The hitting (or first-passage) time quasi-metric H(u, v) from u ∈ V to v ∈ V is the expected number of steps (edges) for a random walk on G beginning at u to reach v for the first time; it is 0 for u = v. This quasi-metric is a weightable quasi-semimetric (cf. Chap. 1).

  The commuting time metric is C(u, v) = H(u, v) + H(v, u).

  Then $C(u,v) = 2\vert E\vert \Omega (u,v)$, where $\Omega (u,v)$ is the resistance metric (or effective resistance), i.e., 0 if u = v and, otherwise, $\frac{1} {\Omega (u,v)}$ is the current flowing into v, when grounding v and applying a 1 volt potential to u (each edge is seen as a resistor of 1 ohm). Also, $\Omega (u,v) =\sup _{f:V \rightarrow \mathbb{R},\,D(f)>0}\frac{(f(u)-f(v))^{2}} {\mathit{DE}(f)}$, where DE(f) is the Dirichlet energy of f, i.e., ∑ _st ∈ E (f(s) − f(t))².

  The above setting can be generalized to weighted digraphs D = (V, E) with arc-weights c _ij for ij ∈ E and the cost of a directed (u − v) path being the sum of the weights of its arcs. Consider the random walk on D, where at each step the walk moves by arc ij with reference probability p _ij proportional to $\frac{1} {c_{\mathit{ij}}}$; set p _ij  = 0 if $\mathit{ij}\notin E$. Saerens et al., 2008, defined the randomized et al. shortest path quasi-distance d(u, v) on vertices of D as the minimum expected cost of a directed (u − v) path in the probability distribution minimizing the expected cost among all distributions having a fixed Kullback–Leibler distance (cf. Chap. 14) with reference probability distribution. In fact, their biased random walk model depends on a parameter θ ≥ 0. For θ = 0 and large θ, the distance d(u, v) + d(v, u) become a metric; it is proportional to the commuting time and the usual path metric, respectively.

• Chebotarev–Shamis metric

  Given α > 0 and a connected weighted multigraph G = (V, E; w) with positive edge-weight function w = (w(e)) _e ∈ E , denote by L = ((l _ij )) the Laplacian (or Kirchhoff) matrix of G, i.e., l _ij  = −w(ij) for i ≠ j and $l_{\mathit{ii}} =\sum _{j\neq i}w(\mathit{ij})$. The Chebotarev–Shamis metric d _α (u, v) (Chebotarev and Shamis, 2000, called $\frac{1} {2}d_{\alpha }(u,v)$ α -forest metric) between vertices u and v is defined by

  $$\displaystyle{2q_{\mathit{uv}} - q_{\mathit{uu}} - q_{\mathit{vv}}}$$

  for the protometric ((g _ij )) = −(I +α L)⁻¹, where I is the identity matrix.

  Chebotarev and Shamis showed that their metric of G = (V, E; w) is the resistance metric of another weighted multigraph, G ^′ = (V ^′, E ^′; w ^′), where V ^′ = V ∪{ 0}, E ^′ = E ∪{ u0: u ∈ V }, while w ^′(e) = α w(e) for all e ∈ E and w ^′(u0) = 1 for all u ∈ V. In fact, there is a bijection between the forests of G and trees of G ^′. This metric becomes the resistance metric of G = (V, E; w) as α → ∞.

  Their forest metric (1997) is the case α = 1 of the α-forest metric.

  Chebotarev, 2010, remarked that $2\ln q_{\mathit{uv}} -\ln q_{\mathit{uu}} -\ln q_{\mathit{vv}}$ is a cutpoint additive metric d _α ^{′ ′}(u, v), i.e., $d_{\alpha }^{{\prime\prime}}(u,w) + d_{\alpha }^{{\prime\prime}}(w,v) = d_{\alpha }^{{\prime\prime}}(u,v)$ holds if and only if w lies on any path connecting u and v. The metric d _α ^{′ ′} is the path metric if α → 0⁺ and the resistance metric if α → ∞.

• Truncated metric

  The truncated metric is a metric on the vertex-set of a graph, which is equal to 1 for any two adjacent vertices, and is equal to 2 for any nonadjacent different vertices. It is the 2-truncated metric for the path metric of the graph. It is the (1, 2) − B -metric if the degree of any vertex is at most B.

• Hsu–Lyuu–Flandrin–Li distance

  Given an m-connected graph G = (V, E) and two vertices u, v ∈ V, a container C(u,v) of width m is a set of m (u − v) paths with any two of them intersecting only in u and v. The length of a container is the length of the longest path in it.

  The Hsu–Lyuu–Flandrin–Li distance between vertices u and v (Hsu–Lyuu, 1991, and Flandrin–Li, 1994) is the minimum of container lengths taken over all containers C(u, v) of width m. This generalization of the path metric is used in parallel architectures for interconnection networks.

• Multiply-sure distance

  The multiply-sure distance is a distance on the vertex-set V of an m-connected weighted graph G = (V, E), defined, for any u, v ∈ V, as the minimum weighted sum of lengths of m disjoint (u − v) paths. This generalization of the path metric helps when several disjoint paths between two points are needed, for example, in communication networks, where m − 1 of (u − v) paths are used to code the message sent by the remaining (u − v) path (see [McCa97]).

• Cut semimetric

  A cut is a partition of a set into two parts. Given a subset S of $V _{n} =\{ 1,\ldots,n\}$, we obtain the partition {S, V _n ∖S} of V _n . The cut semimetric (or split semimetric ) δ _S defined by this partition, is a semimetric on V _n defined by

  $$\displaystyle{\delta _{S}(i,j) = \left \{\begin{array}{ccc} 1,&\mbox{ if }&i\neq j,\vert S \cap \{ i,j\}\vert = 1,\\ 0, & & \mbox{ otherwise}.\end{array} \right.}$$

  Usually, it is considered as a vector in $\mathbb{R}^{\vert E_{n}\vert }$, E(n) = {{ i, j}: 1 ≤ i < j ≤ n}.

  A circular cut of V _n is defined by a subset $S_{[k+1,l]} =\{ k + 1,\ldots,l\}(\mathrm{mod}\,n) \subset V _{n}$: if we consider the points $\{1,\ldots,n\}$ as being ordered along a circle in that circular order, then S _[k+1, l] is the set of its consecutive vertices from k + 1 to l. For a circular cut, the corresponding cut semimetric is called a circular cut semimetric .

  An even cut semimetric (odd cut semimetric ) is δ _S on V _n with even (odd, respectively) | S | . A k -uniform cut semimetric is δ _S on V _n with | S | ∈ { k, n − k}. An equicut semimetric (inequicut semimetric) is δ _S on V _n with $\vert S\vert \in \{\lfloor \frac{n} {2} \rfloor,\lceil \frac{n} {2} \rceil \}$ ($\vert S\vert \notin \{\lfloor \frac{n} {2} \rfloor,\lceil \frac{n} {2} \rceil \}$, respectively); see, for example, [DeLa97].

• Decomposable semimetric

  A decomposable semimetric is a semimetric on $V _{n} =\{ 1,\ldots,n\}$ which can be represented as a nonnegative linear combination of cut semimetrics. The set of all decomposable semimetrics on V _n is a convex cone, called the cut cone CUT _n .

  A semimetric on V _n is decomposable if and only if it is a finite l ₁ -semimetric.

  A circular decomposable semimetric is a semimetric on $V _{n} =\{ 1,\ldots,n\}$ which can be represented as a nonnegative linear combination of circular cut semimetrics. A semimetric on V _n is circular decomposable if and only if it is a Kalmanson semimetric with respect to the same ordering (see [ChFi98]).

• Finite l _p -semimetric

  A finite l _p -semimetric d is a semimetric on $V _{n} =\{ 1,\ldots,n\}$ such that (V _n , d) is a semimetric subspace of the l _p ^m -space $(\mathbb{R}^{m},d_{l_{p}})$ for some $m \in \mathbb{N}$.

  If, instead of V _n , is taken X = { 0, 1}ⁿ, the metric space (X, d) is called the l _p ⁿ -cube. The l ₁ ⁿ-cube is called a Hamming cube; cf. Chap. 4. It is the graphic metric space associated with a hypercube graph H(n, 2), and any subspace of it is called a partial cube.

• Kalmanson semimetric

  A Kalmanson semimetric d with respect to the ordering $1,\ldots,n$ is a semimetric on $V _{n} =\{ 1,\ldots,n\}$ which satisfies the condition

  $$\displaystyle{\max \{d(i,j) + d(r,s),d(i,s) + d(j,r)\} \leq d(i,r) + d(j,s)}$$

  for all 1 ≤ i ≤ j ≤ r ≤ s ≤ n.

  Equivalently, if the points $\{1,\ldots,n\}$ are ordered along a circle C _n in that circular order, then the distance d on V _n is a Kalmanson semimetric if the inequality

  $$\displaystyle{d(i,r) + d(j,s) \leq d(i,j) + d(r,s)}$$

  holds for i, j, r, s ∈ V _n whenever the segments [i, j], [r, s] are crossing chords of C _n .

  A tree-like metric is a Kalmanson metric for some ordering of the vertices of the tree. The Euclidean metric, restricted to the points that form a convex polygon in the plane, is a Kalmanson metric.

• Multicut semimetric

  Let $\{S_{1},\ldots,S_{q}\}$, q ≥ 2, be a partition of the set $V _{n} =\{ 1,\ldots,n\}$, i.e., a collection $S_{1},\ldots,S_{q}$ of pairwise disjoint subsets of V _n such that $S_{1} \cup \ldots \cup S_{q} = V _{n}$.

  The multicut semimetric $\delta _{S_{1},\ldots,S_{q}}$ is a semimetric on V _n defined by

  $$\displaystyle{\delta _{S_{1},\ldots,S_{q}}(i,j) = \left \{\begin{array}{ccc} 0,&\mbox{ if }&i,j \in S_{h}\mbox{ for some }h,1 \leq h \leq q, \\ 1,& & \mbox{ otherwise}.\end{array} \right.}$$

• Oriented cut quasi-semimetric

  Given a subset S of $V _{n} =\{ 1,\ldots,n\}$, the oriented cut quasi-semimetric $\delta _{S}^{^{{\prime}} }$ is a quasi-semimetric on V _n defined by

  $$\displaystyle{\delta _{S}^{^{{\prime}} }(i,j) = \left \{\begin{array}{ccc} 1,&\mbox{ if }& i \in S,j\not\in S,\\ 0, & &\mbox{ otherwise}.\end{array} \right.}$$

  Usually, it is considered as the vector of $\mathbb{R}^{\vert I_{n}\vert }$, $I(n) =\{ (i,j): 1 \leq i\neq j \leq n\}$. The cut semimetric δ _S is $\delta _{S}^{^{{\prime}} } +\delta _{ V _{n}\setminus S}^{^{{\prime}} }$.

• Oriented multicut quasi-semimetric

  Given a partition $\{S_{1},\ldots,S_{q}\}$, q ≥ 2, of V _n , the oriented multicut quasi-semimetric $\delta _{S_{1},\ldots,S_{q}}^{^{{\prime}} }$ is a quasi-semimetric on V _n defined by

  $$\displaystyle{\delta _{S_{1},\ldots,S_{n}}^{^{{\prime}} }(i,j) = \left \{\begin{array}{ccc} 1,& \mbox{ if } &i \in S_{h},j \in S_{m},h < m, \\ 0,&\mbox{ otherwise }&.\end{array} \right.}$$

2 Distance-Defined Graphs

Below we first give some graphs defined in terms of distances between their vertices. Then some graphs associated with metric spaces are presented.

A graph (V, E) is, say, distance-invariant or distance monotone if its metric space (V, d _path) is distance invariant or distance monotone, respectively (cf. Chap. 1). The definitions of such graphs, being straightforward subcases of corresponding metric spaces, will be not given below.

• k -Power of a graph

  The k -power of a graph G = (V, E) is the supergraph G ^k = (V, E ^′) of G with edges between all pairs of vertices having path distance at most k.

• Distance-residual subgraph

  For a connected finite graph G = (V, E) and a set M ⊂ V of its vertices, a distance-residual subgraph is (Luksic and Pisanski, 2010) a subgraph induced on the set of vertices u of G at the maximal point-set distance min _v ∈ M d _path(u, v) from M. Such a subgraph is called vertex-residual if M consists of a vertex, and edge-residual if M consists of two adjacent vertices.

• Isometric subgraph

  A subgraph H of a graph G = (V, E) is called an isometric subgraph if the path metric between any two points of H is the same as their path metric in G.

  A subgraph H is called a convex subgraph if it is isometric, and for any u, v ∈ H every vertex on a shortest (u − v) path belonging to H also belongs to H.

  A subset M ⊂ V is called gated if for every u ∈ V ∖ M there exists a unique vertex g ∈ M (called a gate) lying on a shortest (u − v) path for every v ∈ M. The subgraph induced by a gated set is a convex subgraph.

• Retract subgraph

  A subgraph H of G is called a retract subgraph if it is induced by an idempotent metric mapping of G into itself, i.e., f ² = f: V → V with $d_{\mathrm{path}}(f(u),f(v)) \leq d_{\mathrm{path}}(u,v)$ for u, v ∈ V. Any retract subgraph is isometric.

• Partial cube

  A partial cube is an isometric subgraph of a Hamming cube, i.e., of a hypercube H(m, 2). Similar topological notion was introduced by Acharya, 1983: any graph (V, E) admits a set-indexing f: V ∪ E → 2^X with injective f |  _V , f |  _R and $f(\mathit{uv}) = f(u)\Delta f(v)$ for any (uv) ∈ E. The set-indexing number is min | X | .

• Median graph

  A connected graph G = (V, E) is called a median graph if, for any three vertices u, v, w ∈ V, there exists a unique vertex that lies simultaneously on a shortest (u − v), (u − w) and (w − v) paths, i.e., (V, d _path) is a median metric space.

  The median graphs are exactly retract subgraphs of hypercubes. Also, they are exactly partial cubes such that the vertex-set of any convex subgraph is gated (cf. isometric subgraph).

• Geodetic graph

  A graph is called geodetic if there exists at most one shortest path between any two of its vertices. A graph is called strongly geodetic if there exists at most one path of length less than or equal to the diameter between any two of its vertices.

  A uniformly geodetic graph is a connected graph such that the number of shortest paths between any two vertices u and v depends only on d(u, v).

  A graph is a forest (disjoint union of trees) if and only if there exists at most one path between any two of its vertices.

  The geodetic number of a finite connected graph (V, E) [BuHa90] is min | M | over sets M ⊂ V such that any x ∈ V lies on a shortest (u − v) path with u, v ∈ M.

• k -geodetically connected graph

  A k-connected graph is called (Entringer–Jackson–Slater, 1977) k -geodetically connected (k −GC) if the removal of less than k vertices (or, equivalently, edges) does not affect the path metric between any pair of the remaining vertices.

  2 − GC graphs are called self-repairing. Cf. Hsu–Lyuu–Flandrin–Li distance.

• Interval distance monotone graph

  A connected graph G = (V, E) is called interval distance monotone if any of its intervals I _G (u, v) induces a distance monotone graph, i.e., its path metric is distance monotone, cf. Chap. 1.

  A graph is interval distance monotone if and only if (Zhang–Wang, 2007) each of its intervals is isomorphic to either a path, a cycle or a hypercube.

• Distance-regular graph

  A connected regular (i.e., every vertex has the same degree) graph G = (V, E) of diameter T is called distance-regular (or drg) if, for every two its vertices u, v and any integers 0 ≤ i, j ≤ T, the number $\vert \{w \in V: d_{\mathrm{path}}(u,w)\,=\,i,d_{\mathrm{path}}(v,w) = j\}\vert$ depends only on i, j and k = d _path(u, v), but not on the choice of u and v.

  A special case of it is a distance-transitive graph , i.e., such that its group of automorphisms is transitive, for any 0 ≤ i ≤ T, on the pairs of vertices (u, v) with d _path(u, v) = i. An analog of drg is an edge-regular graph (Fiol–Carriga, 2001).

  Any drg is a distance-balanced graph (or dbg), i.e., | W _u, v  |  =  | W _v, u  | , where W _u, v  = { x ∈ V: d(x, u) < d(x, v)}. Such graph is also called self-median since it is exactly one, metric median (cf. eccentricity in Chap. 1) of which is V. A gbg is called nicely distance-balanced if | W _u, v  | is the same for all edges uv.

  Any drg is a distance degree-regular graph (i.e., | {x ∈ V: d(x, u) = i} | depends only on i; such graph is also called strongly distance-balanced), and a walk-regular graph (i.e., the number of closed walks of length i starting at u depends only on i). van Dam–Omidi, 2013, call a graph strongly walk-regular if there is an l ≥ 2 such that the number of walks of length l from u to v depends only on whether the d(u, v) is 0, 1, or ≥ 2; for l = 2, it is a strongly regular graph, i.e., a drg of diameter 2. A d-Deza graph (Gu, 2013) is a regular graph (V, E) in which there are exactly d different values of | {w ∈ V: d(u, w) = d(v, w) = 1} | for distinct u, v ∈ V.

  A graph G is a distance-regularized graph if for each u ∈ V, if admits an intersection array at vertex u, i.e., the numbers a _i (u) =  | G _i (u) ∩ G ₁(v) | , b _i (u) =  | G _i+1(u) ∩ G ₁(v) | and c _i (u) =  | G _i−1(v) ∩ G ₁(v) | depend only on the distance d(u, v) = i and are independent of the choice of the vertex v ∈ G _i (u). Here, for any i, G _i (w) is the set of all vertices at the distance i from w. Godsil–Shawe-Taylor, 1987, defined such graph and proved that it is either drg or distance-biregular (a bipartite one with vertices in the same class having the same intersection array).

  A drg is also called a metric association scheme or P-polynomial association scheme. A finite polynomial metric space (cf. Chap. 1) is a special case of it, also called a (P and Q)-polynomial association scheme.

• Distance-regular digraph

  A strongly connected digraph D = (V, E) is called distance-regular (Damerell, 1981) if, for any its vertices u, v with d _path(u, v) = k and for any integer 0 ≤ i ≤ k + 1, the number of vertices w, such that d _path(u, w) = i and d _path(v, w) = 1, depends only on k and i, but not on the choice of u and v. In order to find interesting classes of distance-regular digraphs with unbounded diameter, the above definition was weakened by two teams in different directions.

  Call $\overline{d(x,y)} = (d(x,y),d(y,x))$ the two-way distance in digraph D. A strongly connected digraph D = (V, E) is called weakly distance-regular (Wang and Suzuku, 2003) if, for any its vertices u, v with $\overline{d(u,v)} = (k_{1},k_{2})$, the number of vertices w, such that $\overline{d(w,u)} = (i_{1},i_{2})$ and $\overline{d(w,v)} = (j_{1},j_{2})$, depends only on the values k ₁, k ₂, i ₁, i ₂, j ₁, j ₂. Comellas et al., 2004, defined a weakly distance-regular digraph as one in which, for any vertices u and v, the number of u → v walks of every given length only depends on the distance d(u, v).

• Metrically almost transitive graph

  An automorphism of a graph G = (V, E) is a map g: V → V such that u is adjacent to v if and only if g(u) is adjacent to g(v), for any u, v ∈ V. The set Aut(G) of automorphisms of G is a group with respect to the composition of functions.

  A graph G is metrically almost transitive (Krön–Möller, 2008) if there is an integer r such that, for any vertex u ∈ V it holds

  $$\displaystyle{\cup _{g\in \mathit{Aut}(G)}\{g(\overline{B}(u,r) =\{ v \in V: d_{\mathrm{path}}(u,v) \leq r\})\} = V.}$$

• Metric end

  Given an infinite graph G = (V, E), a ray is a sequence (x ₀, x ₁, …) of distinct vertices such that x _i and x _i+1 are adjacent for i ≥ 0.

  Two rays R ₁ and R ₂ are equivalent whenever it is impossible to find a bounded set of vertices F such that any path from R ₁ to R ₂ contains an element of F.

  Metric ends are defined as equivalence classes of metric rays which are rays without infinite, bounded subsets.

• Graph of polynomial growth

  Let G = (V, E) be a transitive locally finite graph. For a vertex v ∈ V, the growth function is defined by

  $$\displaystyle{f(n) = \vert \{u \in V: d(u,v) \leq n\}\vert,}$$

  and it does not depend on v. Cf. growth rate of metric space in Chap. 1.

  The graph G is a graph of polynomial growth if there are some positive constants k, C such that f(n) ≤ Cn ^k for all n ≥ 0. It is a graph of exponential growth if there is a constant C > 1 such that f(n) > C ⁿ for all n ≥ 0.

  A group with a finite symmetric set of generators has polynomial growth rate if the corresponding Cayley graph has polynomial growth. Here the metric ball consists of all elements of the group which can be expressed as products of at most n generators, i.e., it is a closed ball centered in the identity in the word metric, cf. Chap. 10.

• Distance-polynomial graph

  Given a connected graph G = (V, E) of diameter T, for any 2 ≤ i ≤ T denote by G _i the graph $(V,E^{^{{\prime}} })$ with $E^{^{{\prime}} } =\{ e = \mathit{uv} \in E: d_{\mathrm{path}}(u,v) = i\}$. The graph G is called a distance-polynomial if the adjacency matrix of any G _i , 2 ≤ i ≤ T, is a polynomial in terms of the adjacency matrix of G.

  Any distance-regular graph is a distance-polynomial.

• Distance-hereditary graph

  A connected graph is called distance-hereditary (Howorka, 1977) if each of its connected induced subgraphs is isometric.

  A graph is distance-hereditary if each of its induced paths is isometric. A graph is distance-hereditary, bipartite distance-hereditary, block graph, tree if and only if its path metric is a relaxed tree-like metric for edge-weights being, respectively, nonzero half-integers, nonzero integers, positive half-integers, positive integers.

  A graph is called a parity graph if, for any u, v ∈ V, the lengths of all induced (u − v) paths have the same parity. A graph is a parity graph (moreover, distance-hereditary) if and only if every induced subgraph of odd (moreover, any) order of at least five has an even number of Hamiltonian cycles (McKee, 2008).

• Distance magic graph

  A graph G = (V, E) is called a distance magic graph if it admits a distance magic labeling, i.e., a magic constant k > 0 and a bijection $f: V \rightarrow \{ 1,2,\ldots,\vert V \vert \}$ with ∑ _uv ∈ E f(v) = k for every u ∈ V. Introduced by Vilfred, 1994, these graphs generalize magic squares (such complete n-partite graphs with parts of size n).

  Among trees, cycles and K _n , only P ₁, P ₃, C ₄ are distance magic. The hypercube graph H(m, 2) is distance magic if m = 2, 6 but not if m ≡ 0, 1, 3 (mod 4).

• Block graph

  A graph is called a block graph if each of its blocks (i.e., a maximal 2-connected induced subgraph) is a complete graph. Any tree is a block graph.

  A graph is a block graph if and only if its path metric is a tree-like metric or, equivalently, satisfies the four-point inequality.

• Ptolemaic graph

  A graph is called Ptolemaic if its path metric satisfies the Ptolemaic inequality

  $$\displaystyle{d(x,y)d(u,z) \leq d(x,u)d(y,z) + d(x,z)d(y,u).}$$

  A graph is Ptolemaic if and only if it is distance-hereditary and chordal, i.e., every cycle of length greater than 3 has a chord. So, any block graph is Ptolemaic.

• k -cocomparability graph

  A graph G = (V, E) is called (Chang–Ho–Ko, 2003) k -cocomparability graph if its vertex-set admits a linear ordering < such that for any three vertices u < v < w, d(u, w) ≤ k implies d(u, v) ≤ k or d(v, w) ≤ k.

• Distance-perfect graph

  Cvetković et al., 2007, observed that any graph of diameter T has at most k + T ^k vertices, where k is its location number (cf. Chap. 1), i.e., the minimal cardinality of a set of vertices, the path distances from which uniquely determines any vertex. They called a graph distance-perfect if it meets this upper bound and proved that such a graph has T ≠ 2.

• t -irredundant set

  A set S ⊂ V of vertices in a connected graph G = (V, E) is called t -irredundant (Hattingh–Henning, 1994) if for any u ∈ S there exists a vertex v ∈ V such that, for the path metric d _path of G, it holds

  $$\displaystyle{d_{\mathrm{path}}(v,x) \leq t < d_{\mathrm{path}}(v,V \setminus S) =\min _{u\notin S}d_{\mathrm{path}}(v,u).}$$

  The t -irredundance number ir _t of G is the smallest cardinality | S | such that S is t-irredundant but S ∪{ v} is not, for every v ∈ V ∖S.

  The t-domination number γ _t and t-independent number α _t of G are, respectively, the cardinality of the smallest (t + 1)-covering (by the open balls of the radius r + 1) and largest $\lceil \frac{t} {2}\rceil$ -packing of the metric space (V, d _path(u, v)); cf. the radii of metric space in Chap. 1. Then it holds that $\frac{\gamma _{t}+1} {2} \leq \mathit{ir}_{t} \leq \gamma _{t} \leq \alpha _{t}$.

  Let B _S denote {v ∈ V: d(v, S) = 1}. Then $\max _{S\subset V }\vert B_{S}\vert = \vert V \vert -\gamma _{1}$ and $\max _{S\subset V }(\vert B_{S}\vert -\vert S\vert )$ are called the enclaveless number and the differential of G.

• r -Locating-dominating set

  Let D = (V, E) be a digraph and C ⊂ V, and let B _r ⁻(v) denote the set of all vertices x such that there exists a directed (x − v) path with at most r arcs.

  If B _r ⁻(v) ∩ C, v ∈ V ∖ C (respectively, v ∈ V ), are nonempty distinct sets, C is called (Slater, 1984) an r -locating-dominating set (respectively, an r -identifying code; cf. Chap. 16) of D. Such sets of smallest cardinality are called optimal.

• Locating chromatic number

  The locating chromatic number of a graph G = (V, E) is the minimum number of color classes $C_{1},\ldots,C_{t}$ needed to color vertices of G so that any two adjacent vertices have distinct colors and each vertex u ∈ V has distinct color code $(\min _{v\in C_{1}}d(u,v),\ldots,\min _{v\in C_{k}}d(u,v))$.

• k -Distant chromatic number

  The k -distant chromatic number of a graph G = (V, E) is the minimum number of colors needed to color vertices of G so that any two vertices at distance at most k have distinct colors, i.e., it is the chromatic number of the k -power of G.

• Distance between edges

  The distance between edges in a connected graph G = (X, E) is the number of vertices in a shortest path between them. So, adjacent edges have distance 1.

  A distance- k matching of G is a set of edges no two of which are within distance k. For k = 1, it is the usual matching. For k = 2, it is also induced (or strong) matching. A distance-k matching of G is equivalent to an independent set in the k -power of the line graph of G. A distance- k edge-coloring of G is an edge-coloring such that each color class induces a distance-k matching.

  The distance- k chromatic index μ _k (G) is the least integer t such that there exists a distance-t edge-coloring of G. The distance- k matching number ν _k (G) is the largest integer t such that there exists a distance-t matching in G with t edges. It holds that μ _k (G)ν _k (G) ≥ | E | .

  The distance between faces of a plane graph is the number of vertices in a shortest path between them. A distance- k face-coloring is a face-coloring such that any two faces at distance at most k have different colors. The distance- k face chromatic index is the least integer t such that such coloring exists.

• Rainbow distance

  In an edge-colored graph, the rainbow distance is (Chartrand and Zhang, 2005) the length of a shortest rainbow (i.e., containing no color twice) path.

  In a vertex-colored graph, the colored distance is (Dankelmann et al., 2001) the sum of distances between all unordered pairs of vertices having different colors.

• D -distance graph

  Given a set D of positive numbers containing 1 and a metric space (X, d), the D -distance graph is a graph G = (V = X, E) with the edge-set E = { uv: d(u, v) ∈ D} (cf. D-chromatic number in Chap. 1). If (X, d) is path metric of a graph H, then G is called the distance power H ^D of H.

  Alon–Kupavsky, 2014, call G (in the case $(X,d) = \mathbb{E}^{n}$, d = { 1}) the faithful unit-distance graph, using term unit-distance graph for E ⊆ { (u, v):  | | u − v | | ₂ = 1}.

  For a positive number t, the signed distance graph is (Fiedler, 1969) a signed graph with the vertex-set X in which vertices x, y are joined by a positive edge if t > d(x, y), by a negative edge if d(x, y) > t, and not joined if d(x, y) = t.

  A D-distance graph is called a distance graph (or unit-distance graph) if D = { 1}, an ε-unit graph if D = [1 −ε, 1 +ε], a unit-neighborhood graph if D = (0, 1], an integral-distance graph if $D = \mathbb{Z}_{+}$, a rational-distance graph if $D = \mathbb{Q}_{+}$, and a prime-distance graph if D is the set of prime numbers (with 1).

  Every finite graph can be represented by a D-distance graph in some $\mathbb{E}^{n}$. The minimum dimension of such a Euclidean space is called the D-dimension of G. A matchstick graph is a crossingless unit-distance graph in $\mathbb{E}^{2}$.

• Distance-number of a graph

  Given a graph G = (V, E), its degenerate drawing is a mapping $f: V \rightarrow \mathbb{R}^{2}$ such that | f(V ) |  =  | V | and f(uv) is an open straight-line segment joining the vertices f(u) and f(v) for any edge uv ∈ E; it is a drawing if, moreover, $f(w)\notin f(\mathit{uv})$ for any uv ∈ E and w ∈ V.

  The distance-number dn(G) of a graph G is (Carmi et al., 2008) the minimum number of distinct edge-lengths in a drawing of G.

  The degenerate distance-number of G, denoted by ddn(G), is the minimum number of distinct edge-lengths in a degenerated drawing of G. The first of the Erdös-type distance problems in Chap. 19 is equivalent to determining ddn(K _n ).

• Dimension of a graph

  The dimension dim(G) of a graph G is (Erdös–Harary–Tutte, 1965) the minimum k such that G has a unit-distance representation in $\mathbb{R}^{k}$, i.e., every edge is of length 1. The vertices are mapped to distinct points of $\mathbb{R}^{k}$, but edges may cross.

  For example, dim(G) = n − 1, 4, 2 for G = K _n , K _m, n , C _n (m ≥ n ≥ 3).

• Bar-and-joint framework

  A n-dimensional bar-and-joint framework is a pair (G, f), where G = (V, E) is a finite graph (no loops and multiple edges) and $f: V \rightarrow \mathbb{R}^{n}$ is a map with f(u) ≠ f(v) whenever uv ∈ E. The framework is a straight line realization of G in $\mathbb{R}^{n}$ in which the length of an edge uv ∈ E is given by | | f(u) − f(v) | | ₂.

  The vertices and edges are called joints and bars, respectively, in terms of Structural Engineering. A tensegrity structure (Fuller, 1948) is a mechanically stable bar framework in which bars are either cables (tension elements which cannot get further apart), or struts (compression elements which cannot get closer together).

  A framework (G, f) is globally rigid if every framework (G, f ^′), satisfying $\vert \vert f(u) - f(v)\vert \vert _{2} = \vert \vert f^{{\prime}}(u) - f^{{\prime}}(v)\vert \vert _{2}$ for all uv ∈ E, also satisfy it for all u, v ∈ V. A framework (G, f) is rigid if every continuous motion of its vertices which preserves the lengths of all edges, also preserves the distances between all pairs of vertices. The framework (G, f) is generic if the set containing the coordinates of all the points f(v) is algebraically independent over the rationals. The graph G is n-rigid if every its n-dimensional generic realization is rigid. For generic frameworks, rigidity is equivalent to the stronger property of infinitesimal rigidity.

  An infinitesimal motion of (G, f) is a map $m: V \rightarrow \mathbb{R}^{n}$ with (m(u) − m(v))(f(u) − f(v)) = 0 whenever uv ∈ E. A motion is trivial if it can be extended to an isometry of $\mathbb{R}^{n}$. A framework is an infinitesimally rigid if every motion of it is trivial, and it is isostatic if, moreover, the deletion of any its edge will cause loss of rigidity. (G, f) is an elastic framework if, for any ε > 0, there exists a δ > 0 such that for every edge-weighting $w: E \rightarrow \mathbb{R}_{>0}$ with $\max _{\mathit{uv}\in E}\vert w(\mathit{uv}) -\vert \vert f(u) - f(v)\vert \vert _{2}\vert \leq \delta$, there exist a framework (G, f ^′) with $\max _{v\in V }\vert \vert f(u) - f^{{\prime}}(v)\vert \vert _{2} <\epsilon$.

  A framework (G, f) with | | f(u) − f(v) | | ₂ > r if u, v ∈ V, u ≠ c and | | f(u), f(v) | | ₂ ≤ R if uv ∈ E, for some 0 < r < R, is called (Doyle–Snell, 1984) a civilized drawing of a graph. The random walks on such graphs are recurrent if n = 1, 2.

• Distance constrained labeling

  Given a sequence $\alpha = (\alpha _{1},\ldots,\alpha _{k})$ of distance constraints $\alpha _{1} \geq \ldots \geq \alpha _{k} > 0$, a λ _α -labeling of a graph G = (V, E) is an assignment of labels f(v) from the set $\{0,1,\ldots,\lambda \}$ of integers to the vertices v ∈ V such that, for any t with 0 ≤ t ≤ k, | f(v) − f(u) | ≥ α _t whenever the path distance between u and v is t.

  The radio frequency assignment problem, where vertices are transmitters (available channels) and labels represent frequencies of not-interfering channels, consists of minimizing λ. Distance-two labeling is the main interesting case α = (2, 1); its span is the difference between the largest and smallest labels used.

• Distance-related graph embedding

  An embedding of the guest graph G = (V ₁, E ₁) into the host graph H = (V ₂, E ₂) with | V ₁ | ≤ | V ₂ | , is an injective map from V ₁ into V ₂.

  The wire length , dilation and antidilation of G in H are

  $$\displaystyle{\min _{f}\sum _{(\mathit{uv})\in E_{1}}d_{H}(f(u),f(v)),\,\,\min _{f}\max _{(\mathit{uv})\in E_{1}}d_{H}(f(u),f(v)),\,\,\max _{f}\min _{(\mathit{uv})\in E_{1}}d_{H}(f(u),f(v)),}$$

  respectively, where f is any embedding of G into H. The main distance-related graph embedding problems consist of finding or estimating these three parameters.

  The bandwidth and antibandwidth of G is the dilation and antidilation, respectively, of G in a path H with V ₁ vertices.

• Bandwidth of a graph

  Given a graph G = (V, E) with | V |  = n, its ordering is a bijective mapping $f: V \rightarrow \{ 1,\ldots,n\}$. Given a number b > 0, the bandwidth problem for (G, b) is the existence of ordering f with the stretch max _uv ∈ E  | f(u) − f(v) | at most b.

  The bandwidth of G, denoted by bw(G), is the minimum stretch over all f.

  The antibandwidth problem for G is to find ordering f with maximal $\min _{\mathit{uv}\in E}\vert f(u) - f(v)\vert$ (antibandwidth).

• Path distance width of a graph

  Given a connected graph G = (V, E), an ordered partition V = ∪ _i = 1 ^t L _i of its vertices is called a distance structure on G if $L_{i} =\{ v \in V:\min _{u\in L_{1}}d_{\mathrm{path}}(u,v) = i - 1\}$ for 1 ≤ i ≤ t. The structure is rooted if | L ₁ |  = 1.

  The path distance width pwd(G) of G is defined (Yamazaki et al., 1999) as $\min \max _{1\leq i\leq t}\vert L_{i}\vert$ over all distance structures on G.

  An ordered partition V = ∪ _i = 1 ^t L _i is called a level structure on G if for each edge uv with u ∈ L _i and v ∈ L _j , it holds that | i − j | ≤ 1. The level width (or strong pathwidth) lw(G) is minmax_{1 ≤ i ≤ t}  | L _i  | over all level structures.

  Clearly, lw(G) ≤ pdw(G). Yamazaki et al., 1999, proved that pdw(G) can be arbitrarily larger than the bandwidth bw(G) and lw(G) ≤ bw(G) < 2lw(G).

• Tree-length of a graph

  A tree decomposition of a graph G = (V, E) is a pair of a tree T with vertex-set W and a family of subsets {X _i : i ∈ W} of V with ∪ _i ∈ W X _i  = V such that

  • 1. for every edge (uv) ∈ E, there is a subset X _i containing u, v, and

  • 2. for every v ∈ V, the set i ∈ W: v ∈ X _i induces a connected subtree of T.

  The chordal graphs (i.e., ones without induced cycles of length at least 4) are exactly those admitting a tree decomposition where every X _i is a clique.

  For tree decomposition, the tree-length is max _i ∈ W diam(X _i ) (diam(X _i ) is the diameter of the subgraph of G induced by X _i ) and tree-width is max _i ∈ W  | X _i  | − 1. The tree-length of G (Dourisboure–Gavoille, 2004) and its tree-width (Robertson–Seymour, 1986) are the minima, over all tree decompositions, of above tree-length and tree-width. The path-length G is defined taking as trees only paths.

  Given a linear ordering $e_{1},\ldots,e_{\vert E\vert }$ of the edges of G, let, for 1 ≤ i <  | E | , denote by G _≤ i and G _i < the graphs induced by the edges $\{e_{1},\ldots,e_{i}\}$ and $\{e_{i+1},\ldots,e_{\vert E\vert }\}$, respectively. The linear-length is $\max _{1\leq i<\vert E\vert }\mathit{diam}(V (G_{\leq i}) \cap V (G_{i<}))$. The linear-length of G (Umezawa–Yamazaki, 2009) is the minimum of the above linear-length taken over all the linear orderings of its edges.

• Spatial graph

  A spatial graph (or spatial network) is a graph G = (V, E), where each vertex v has a spatial position $(v_{1},\ldots,v_{n}) \in \mathbb{R}^{n}$. (G is called a geometric graph if it is drawn on $\mathbb{R}^{2}$ and its edges are straight-line segments.)

  The graph-theoretic dilation and geometric dilation of G are, respectively:

  $$\displaystyle{\max _{v,u\in V } \frac{d(v,u)} {\vert \vert v - u\vert \vert _{2}}\mbox{ and }\max _{(vu)\in E} \frac{d(v,u)} {\vert \vert v - u\vert \vert _{2}}.}$$

• Distance Geometry problem

  Given a weighted finite graph G = (V, E; w), the Distance Geometry problem (DGP) is the problem of realizing it as a spatial graph G = (V ^′, E ^′), where x: V → V ^′ is a bijection with $x(v) = (v_{1},\ldots,v_{n}) \in \mathbb{R}^{n}$ for every v ∈ V and E ^′ = { (x(u)x(v)): (uv) ∈ E}, so that for every edge (uv) ∈ E it holds that

  $$\displaystyle{\vert \vert x(u) - x(u)\vert \vert _{2} = w(\mathit{uv}).}$$

  The main application of DGP is the molecular DGP: to find the coordinates of the atoms of a given molecular conformation are by exploiting only some of the distances between pairs of atoms found experimentally; cf. [MLLM13].

• Arc routing problems

  Given a finite set X, a quasi-distance d(x, y) on it and a set A ⊆ { (x, y): x, y ∈ X}, consider the weighted digraph D = (X, A) with the vertex-set X and arc-weights d(x, y) for all arcs (x, y) ∈ A. For given sets V of vertices and E of arcs, the arc routing problem consists of finding a shortest (i.e., with minimal sum of weights of its arcs) (V,E)-tour, i.e., a circuit in D = (X, A), visiting each vertex in V and each arc in E exactly once or, in a variation, at least once.

  The Asymmetric Traveling Salesman problem corresponds to the case V = X, E = ∅; the Traveling Salesman problem is the symmetric version of it (usually, each vertex should be visited exactly once). The Bottleneck Traveling Salesman problem consists of finding a (V, E)-tour T with smallest max_{(x, y) ∈ T} d(x, y).

  The Windy Postman problem corresponds to the case V = ∅, E = A, while the Chinese Postman problem is the symmetric version of it.

  The above problems are also considered for general arc- or edge-weights; then, for example, the term Metric TSP is used when edge-weights in the Traveling Salesman problem satisfy the triangle inequality, i.e., d is a quasi-semimetric.

• Steiner distance of a set

  The Steiner distance of a set S ⊂ V of vertices in a connected graph G = (V, E) is (Chartrand et al., 1989) the minimum size (number of edges) of a connected subgraph of G, containing S. Such a subgraph is a tree, and is called a Steiner tree for S. Cf. general Steiner diversity in Steiner ratio (Chap. 1).

  The Steiner distance of the set S = { u, v} is the path metric between u and v. The Steiner k-diameter of G is the maximum Steiner distance of any k-subset of V.

• t -Spanner

  A factor, i.e., a spanning subgraph, H = (V, E(H)) of a connected graph G = (V, E) is called a t -spanner (or t-multiplicative spanner) of G if, for every u, v ∈ V, the inequality d _path ^H(u, v)∕d _path ^G(u, v) ≤ t holds. The value t is called the stretch factor (or dilation) of H. Cf. distance-related graph embedding and spatial graph.

  The graph H = (V, E(H)) is called a k-additive spanner of G if, for every u, v ∈ V, the inequality d _path ^H(u, v) ≤ d _path ^G(u, v) + k holds.

  Mulder and Nebeský, 2012, defined, for connected H, the guide of (H, G) as the ternary relation R ⊂ V × V × V consisting of ordered triples (u, w, v) such that uw ∈ E and $d_{\mathrm{path}}^{H}(u,w) + d_{\mathrm{path}}^{H}(w,v) = d_{\mathrm{path}}^{H}(u,v)$. The guide of (G, G) is called the step ternary relation; cf. metric betweenness in Chap. 1.

• Optimal realization of metric space

  Given a finite metric space (X, d), a realization of it is a weighted graph G = (V, E; w) with X ⊂ V such that d(x, y) = d _G (x, y) holds for all x, y ∈ X.

  The realization is optimal if it has minimal ∑ _{(uv) ∈ E} w(uv).

• Proximity graph

  Given a finite subset V of a metric space (X, d), its proximity graph is a graph representing neighbor relationships between points of V. Such graphs are used in Computational Geometry and many real-world problems. The main examples are presented below. Cf. underlying graph of a metric space in Chap. 1.

  A spanning tree of V is a set T of | V | − 1 unordered pairs (x, y) of different points of V forming a tree on V; the weight of T is ∑ _{(x, y) ∈ T} d(x, y). A minimum spanning tree MST(V ) of V is a spanning tree with the minimal weight. Such a tree is unique if the edge-weights are distinct.

  A nearest neighbor graph is the digraph NNG(V ) = (V, E) with vertex-set $V = v_{1},\ldots,v_{\vert V \vert }$ and, for x, y ∈ V, xy ∈ E if y is the nearest neighbor of x, i.e., $d(x,y) =\min _{v_{i}\in V \setminus \{x\}}d(x,v_{i})$ and only v _i with maximal index i is picked. The k-nearest neighbor graph arises if k such v _i with maximal indices are picked. The undirect version of NNG(V ) is a subgraph of MST(V ).

  A relative neighborhood graph is (Toussaint, 1980) the graph RNG(V ) = (V, E) with vertex-set V and, for x, y ∈ V, xy ∈ E if there is no point z ∈ V with max{d(x, z), d(y, z)} < d(x, y). Also considered, for $(X,d) = (\mathbb{R}^{2},\vert \vert x - y\vert \vert _{2})$, the related Gabriel graph GG(V ) (in general, β-skeleton) and Delaunay triangulation DT(V ); then $\mathit{NNG}(V ) \subseteq \mathit{MST}(V ) \subseteq \mathit{RNG}(V ) \subseteq \mathit{GG}(V ) \subseteq \mathit{DT}(V )$.

  For any x ∈ V, its sphere of influence is the open metric ball B(x, r _x ) = { z ∈ X: d(x, z) < r} in (X, d) centered at x with radius $r_{x} =\min _{z\in V \setminus \{x\}}d(x,z)$.

  Sphere of influence graph is the graph SIG(V ) = (V, E) with vertex-set V and, for x, y ∈ V, xy ∈ E if B(x, r _x ) ∩ B(y, r _y ) ≠ ∅; so, it is a proximity graph and an intersection graph. The closed sphere of influence graph is the graph CSIG(V ) = (V, E) with xy ∈ E if $\overline{B(x,r_{x})} \cap \overline{B(y,r_{y})}\neq \varnothing$.

3 Distances on Graphs

• Chartrand–Kubicki–Schultz distance

  The Chartrand–Kubicki–Schultz distance (or ϕ-distance, 1998) between two connected graphs G ₁ = (V ₁, E ₁) and G ₂ = (V ₂, E ₂) with | V ₁ |  =  | V ₂ |  = n is

  $$\displaystyle{\min \{\sum \vert d_{G_{1}}(u,v) - d_{G_{2}}(\phi (u),\phi (v))\vert \},}$$

  where $d_{G_{1}},d_{G_{2}}$ are the path metrics of graphs G ₁, G ₂, the sum is taken over all unordered pairs u, v of vertices of G ₁, and the minimum is taken over all bijections ϕ: V ₁ → V ₂.

• Subgraph metric

  Let $\mathbb{F} =\{ F_{1} = (V _{1},E_{1}),F_{2} = (V _{2},E_{2}),\ldots,\}$ be the set of isomorphism classes of finite graphs. Given a finite graph G = (V, E), denote by s _i (G) the number of injective homomorphisms from F _i into G, i.e., the number of injections ϕ: V _i  → V with ϕ(x)ϕ(y) ∈ E if xy ∈ E _i divided by the number $\frac{\vert V \vert !} {(\vert V \vert -\vert V _{i}\vert )!}$ of such injections from F _i with | V _i  | ≤ | V | into K _| V |. Set s(G) = (s _i (G)) _i = 1 ^∞ ∈ [0, 1]^∞.

  Let d be the Cantor metric (cf. Chap. 18) $d(x,y) =\sum _{ i=1}^{\infty }2^{-i}\vert x_{i} - y_{i}\vert$ on [0, 1]^∞ or any metric on [0, 1]^∞ inducing the product topology. Then Bollobás–Riordan, 2007, defined the subgraph metric between the graphs G ₁ and G ₂ as

  $$\displaystyle{d(s(G_{1}),s(G_{2}))}$$

  and generalized it on kernels (or graphons), i.e., symmetric measurable functions $k: [0,1] \times [0,1] \rightarrow \mathbb{R}_{\geq 0}$, replacing G by k and the above s _i (G) by

  $$\displaystyle{s_{i}(k) =\int _{[0,1]^{\vert V_{i}\vert }}\prod _{st\in E_{i}}k(x_{s}x_{t})\prod _{s=1}^{\vert V _{i}\vert }\mathit{dx}_{ s}.}$$

• Benjamini–Schramm metric

  The rooted graphs (G, o) and (G ^′, o ^′) (where $G = (V,E),G^{{\prime}} = (V ^{{\prime}},E^{{\prime}})$ and $o \in V,o^{{\prime}}\in V ^{{\prime}}$) are isomorphic is there is a graph-isomorphism of G onto G ^′ taking o to o ^′. Let X be the set of isomorphism classes of rooted connected locally finite graphs and let (G, o), (G ^′, o ^′) be representatives of two classes.

  Let k be the supremum of all radii r, for which rooted metric balls $(\overline{B}_{G}(o,r),o)$ and $(\overline{B}_{G^{{\prime}}}(o^{{\prime}},r),o^{{\prime}})$ (in the usual path metric) are isomorphic as rooted graphs. Benjamini and Schramm, 2001, defined the metric 2^−k between classes represented by (G, o) and (G ^′, o ^′). Here 2^−∞ means 0. Benjamini and Curien, 2011, defined the similar distance $\frac{1} {1+k}$.

• Rectangle distance on weighted graphs

  Let G = G(α, β) be a complete weighted graph on $\{1,\ldots,n\}$ with vertex-weights α _i  > 0, 1 ≤ i ≤ n, and edge-weights $\beta _{\mathit{ij}} \in \mathbb{R}$, 1 ≤ i < j ≤ n. Denote by A(G) the n × n matrix ((a _ij )), where $a_{\mathit{ij}} = \frac{\alpha _{i}\alpha _{j}\beta _{\mathit{ij}}} {(\sum _{1\leq i\leq n}\alpha _{i})^{2}}$.

  The rectangle distance (or cut distance) between two weighted graphs G = G(α, β) and G ^′ = G(α ^′, β ^′) (with vertex-weights (α _i ^′) and edge-weights (β _ij ^′)) is defined (Borgs–Chayes–Lovász–Sós–Vesztergombi, 2007) by

  $$\displaystyle{\max _{I,J\subset \{1,\ldots,n\}}\left \vert \sum _{i\in I,j\in J}(a_{\mathit{ij}} - a_{\mathit{ij}}^{{\prime}})\right \vert +\sum _{ i=1}^{n}\left \vert \frac{\alpha _{i}} {\sum _{1\leq j\leq n}\alpha _{j}} - \frac{\alpha _{i}^{{\prime}}} {\sum _{1\leq j\leq n}\alpha _{j}^{{\prime}}}\right \vert,}$$

  where A(G) = ((a _ij )) and A(G ^′) = ((a _ij ^′)).

  In the case (α _i ^′) = (α _i ), the rectangle distance is | | A(G) − A(G ^′) | |  _cut , i.e., the cut norm metric (cf. Chap. 12) between matrices A(G) and A(G ^′) and the rectangle distance from Frieze–Kannan, 1999. In this case, the l ₁- and l ₂-metrics between two weighted graphs G and G ^′ are defined as | | A(G) − A(G ^′) | | ₁ and | | A(G) − A(G ^′) | | ₂, respectively. The subcase α _i  = 1 for all 1 ≤ i ≤ n corresponds to unweighted vertices. Cf. the Robinson–Foulds weighted metric.

  Authors generalized the rectangle distance on kernels (or graphons), i.e., symmetric measurable functions $k: [0,1] \times [0,1] \rightarrow \mathbb{R}_{\geq 0}$, using the cut norm $\vert \vert k\vert \vert _{\mathit{cut}} =\sup _{S,T\subset [0,1]}\vert \int _{S\times T}k(x,y)\mathit{dxdy}\vert$.

  A map ϕ: [0, 1] → [0, 1] is measure-preserving if, for any measurable subset A ⊂ [0, 1], the measures of A and ϕ ⁻¹(A) are equal. For a kernel k, define the kernel k ^ϕ by k ^ϕ(x, y) = k(ϕ(x), ϕ(y)). The Lovász–Szegedy semimetric (2007) between kernels k ₁ and k ₁ is defined by

  $$\displaystyle{\inf _{\phi }\vert \vert k_{1}^{\phi } - k_{ 2}\vert \vert _{\mathit{cut}},}$$

  where ϕ ranges over all measure-preserving bijections [0, 1] → [0, 1]. Cf. Chartrand–Kubicki–Schultz distance.

• Subgraph-supergraph distances

  A common subgraph of graphs G ₁ and G ₂ is a graph which is isomorphic to induced subgraphs of both G ₁ and G ₂. A common supergraph of graphs G ₁ and G ₂ is a graph which contains induced subgraphs isomorphic to G ₁ and G ₂.

  The Zelinka distance d _Z [Zeli75] on the set G of all graphs (more exactly, on the set of all equivalence classes of isomorphic graphs) is defined by

  $$\displaystyle{d_{Z} =\max \{ n(G_{1}),n(G_{2})\} - n(G_{1},G_{2})}$$

  for any G ₁, G ₂ ∈ G, where n(G _i ) is the number of vertices in G _i , i = 1, 2, and n(G ₁, G ₂) is the maximum number of vertices of their common subgraph.

  The Bunke–Shearer metric (1998) on the set of nonempty graphs is defined by

  $$\displaystyle{1 - \frac{n(G_{1},G_{2})} {\max \{n(G_{1}),n(G_{2})\}}.}$$

  Given any set M of graphs, the common subgraph distance d _M on M is

  $$\displaystyle{\max \{n(G_{1}),n(G_{2})\} - n(G_{1},G_{2}),}$$

  and the common supergraph distance d _M ^∗ is defined, for any G ₁, G ₂ ∈ M, by

  $$\displaystyle{N(G_{1},G_{2}) -\min \{ n(G_{1}),n(G_{2})\},}$$

  where n(G _i ) is the number of vertices in G _i , i = 1, 2, while n(G ₁, G ₂) and N(G ₁, G ₂) are the maximal order of a common subgraph G ∈ M and the minimal order of a common supergraph H ∈ M, respectively, of G ₁ and G ₂.

  d _M is a metric on M if the following condition (i) holds:

  • (i) if H ∈ M is a common supergraph of G ₁, G ₂ ∈ M, then there exists a common subgraph G ∈ M of G ₁ and G ₂ with n(G) ≥ n(G ₁) + n(G ₂) − n(H).

    d _M ^∗ is a metric on M if the following condition (ii) holds:

  • (ii) if G ∈ M is a common subgraph of G ₁, G ₂ ∈ M, then there exists a common supergraph H ∈ M of G ₁ and G ₂ with n(H) ≤ n(G ₁) + n(G ₂) − n(G).

  One has d _M  ≤ d _M ^∗ if the condition (i) holds, and d _M  ≥ d _M ^∗ if (ii) holds.

  The distance d _M is a metric on the set G of all graphs, the set of all cycle-free graphs, the set of all bipartite graphs, and the set of all trees. The distance d _M ^∗ is a metric on the set G of all graphs, the set of all connected graphs, the set of all connected bipartite graphs, and the set of all trees. The Zelinka distance d _Z coincides with d _M and d _M ^∗ on the set G of all graphs. On the set T of all trees the distances d _M and d _M ^∗ are identical, but different from the Zelinka distance.

  The Zelinka distance d _Z is a metric on the set G(n) of all graphs with n vertices, and is equal to n − k or to K − n for all G ₁, G ₂ ∈ G(n), where k is the maximum number of vertices of a common subgraph of G ₁ and G ₂, and K is the minimum number of vertices of a common supergraph of G ₁ and G ₂.

  On the set T(n) of all trees with n vertices the distance d _Z is called the Zelinka tree distance (see, for example, [Zeli75]).

• Fernández–Valiente metric

  Given graphs G and H, let G ₁ = (V ₁, E ₁) and G ₂ = (V ₂, E ₂) be their maximum common subgraph and minimum common supergraph; cf. subgraph-supergraph distances. The Fernández–Valiente metric (2001) between G and H is

  $$\displaystyle{(\vert V _{2}\vert + \vert E_{2}\vert ) - (\vert V _{1}\vert + \vert E_{1}\vert ).}$$

• Graph edit distance

  The graph edit distance (Axenovich–Kézdy–Martin, 2008, and Alon–Stav, 2008) between graphs G and G ^′ on the same labeled vertex-set is defined by

  $$\displaystyle{d_{\mathit{ed}}(G,G^{{\prime}}) = \vert E(G)\Delta E(G^{{\prime}})\vert.}$$

  It is the minimum number of edge deletions or additions needed to transform G into G ^′, and half of the Hamming distance between their adjacency matrices.

  Given a graph property (i.e., a family $\mathcal{H}$ of graphs), let $d_{\mathit{ed}}(G,\mathcal{H})$ be $\min \{d_{\mathit{ed}}(G,G^{{\prime}}): V (G^{{\prime}}) = V (G),G^{{\prime}}\in \mathcal{H}\}$. Given a number p ∈ (0, 1], the edit distance function of a property $\mathcal{H}$ is (if this limit exists) defined by

  $$\displaystyle{\mathit{ed}_{\mathcal{H}}(p)\,=\,\lim _{n\rightarrow \infty }\max \left \{d_{\mathit{ed}}(G,\mathcal{H}): \vert V (G)\vert = n,\vert E(G)\vert \,=\,\left \lfloor p{n\choose 2}\right \rfloor \right \}\left ({n\choose 2}\right )^{-1}.}$$

  If $\mathcal{H}$ is hereditary (closed under the taking induced subgraphs) and nontrivial (contains arbitrarily large graphs), then (Balogh–Martin, 2008) it holds

  $$\displaystyle{\mathit{ed}_{\mathcal{H}}(p) =\lim _{n\rightarrow \infty }\mathbb{E}[d_{\mathit{ed}}(G(n,p),\mathcal{H})]\left ({n\choose 2}\right )^{-1};}$$

  G(n, p) is the random graph (Chap. 1) on n vertices with edge probability p.

  Bunke, 1997, defined the graph edit distance between vertex- and edge-labeled graphs G ₁ and G ₂ as the minimal total cost of matching G ₁ and G ₂, using deletions, additions and substitutions of vertices and edges. Cf. also tree, top-down, unit cost and restricted edit distance between rooted trees.

  The Bayesian graph edit distance between two relational graphs (i.e., triples (V, E, A), where V, E, A are the sets of vertices, edges, vertex-attributes) is (Myers–Wilson–Hancock, 2000) their graph edit distance with costs defined by probabilities of operations along an editing path seen as a memoryless error process. Cf. transduction edit distances (Chap. 11) and Bayesian distance (Chap. 14).

  The structural Hamming distance between two digraphs G = (X, E) and G ^′ = (X, E ^′) is defined (Acid–Campos, 2003) as $\mathit{SHD}(G,G^{{\prime}}) = \vert E\Delta E^{{\prime}}\vert$.

• Edge distance

  The edge distance on the set of all graphs is defined (Baláž et al., 1986) by

  $$\displaystyle{\vert E_{1}\vert + \vert E_{2}\vert - 2\vert E_{12}\vert + \vert \vert V _{1}\vert -\vert V _{2}\vert \vert }$$

  for any graphs G ₁ = (V ₁, E ₁) and G ₂ = (V ₂, E ₂), where G ₁₂ = (V ₁₂, E ₁₂) is a common subgraph of G ₁ and G ₂ with maximal number of edges. This distance has many applications in Organic and Medical Chemistry.

• Contraction distance

  The contraction distance is a distance on the set G(n) of all graphs with n vertices defined by

  $$\displaystyle{n - k}$$

  for any G ₁, G ₂ ∈ G(n), where k is the maximum number of vertices of a graph which is isomorphic simultaneously to a graph, obtained from each of G ₁ and G ₂ by a finite number of edge contractions. To perform the contraction of the edge uv ∈ E of a graph G = (V, E) means to replace u and v by one vertex that is adjacent to all vertices of V ∖{u, v} which were adjacent to u or to v.

• Edge move distance

  The edge move distance (Baláž et al., 1986) is a metric on the set G(n, m) of all graphs with n vertices and m edges, defined, for any G ₁, G ₂ ∈ G(m, n), as the minimum number of edge moves necessary for transforming the graph G ₁ into the graph G ₂. It is equal to m − k, where k is the maximum size of a common subgraph of G ₁ and G ₂.

  An edge move is one of the edge transformations, defined as follows: H can be obtained from G by an edge move if there exist (not necessarily distinct) vertices u, v, w, and x in G such that uv ∈ E(G), $\mathit{wx}\notin E(G)$, and H = G −uv + wx.

• Edge jump distance

  The edge jump distance is an extended metric (which in general can take the value ∞) on the set G(n, m) of all graphs with n vertices and m edges defined, for any G ₁, G ₂ ∈ G(m, n), as the minimum number of edge jumps necessary for transforming G ₁ into G ₂.

  An edge jump is one of the edge transformations, defined as follows: H can be obtained from G by an edge jump if there exist four distinct vertices u, v, w, and x in G, such that uv ∈ E(G), $\mathit{wx}\notin E(G)$, and H = G −av + wx.

• Edge flipping distance

  Let P = { v ₁, …, v _n } be a collection of points on the plane. A triangulation T of P is a partition of the convex hull of P into a set of triangles such that each triangle has a disjoint interior and the vertices of each triangle are points of P.

  The edge flipping distance is a distance on the set of all triangulations of P defined, for any triangulations T and T ₁, as the minimum number of edge flippings necessary for transforming T into T ₁.

  An edge e of T is called flippable if it is the boundary of two triangles t and $t^{^{{\prime}} }$ of T, and $C = t \cup t^{^{{\prime}} }$ is a convex quadrilateral. The flipping e is one of the edge transformations, which consists of removing e and replacing it by the other diagonal of C. Edge flipping is an special case of edge jump.

  The edge flipping distance can be extended on pseudo-triangulations, i.e., partitions of the convex hull of P into a set of disjoint interior pseudo-triangles (simply connected subsets of the plane that lie between any three mutually tangent convex sets) whose vertices are given points.

• Edge rotation distance

  The edge rotation distance (Chartand–Saba–Zou, 1985) is a metric on the set G(n, m) of graphs with n vertices and m edges, defined, for any G ₁, G ₂, as the minimum number of edge rotations needed for transforming G ₁ into G ₂.

  An edge rotation is one of the edge transformations, defined as follows: H can be obtained from G by an edge rotation if there exist distinct vertices u, v, and w in G, such that uv ∈ E(G), $\mathit{uw}\notin E(G)$, and H = G −uv + uw.

• Tree edge rotation distance

  The tree edge rotation distance is a metric on the set T(n) of all trees with n vertices defined, for all T ₁, T ₂ ∈ T(n), as the minimum number of tree edge rotations necessary for transforming T ₁ into T ₂. A tree edge rotation is an edge rotation performed on a tree, and resulting in a tree.

  For T(n) the tree edge rotation and the edge rotation distances may differ.

• Edge shift distance

  The edge shift distance (or edge slide distance ) is a metric (Johnson, 1985) on the set G _c (n, m) of all connected graphs with n vertices and m edges defined, for any G ₁, G ₂ ∈ G _c (m, n), as the minimum number of edge shifts necessary for transforming G ₁ into G ₂.

  An edge shift is one of the edge transformations, defined as follows: H can be obtained from G by an edge shift if there exist distinct vertices u, v, and w in G such that uv, vw ∈ E(G), $\mathit{uw}\notin E(G)$, and H = G −uv + uw. Edge shift is a special kind of edge rotation in the case when the vertices v, w are adjacent in G.

  The edge shift distance can be defined between any graphs G and H with components G _i (1 ≤ i ≤ k) and H _i (1 ≤ i ≤ k), respectively, such that G _i and H _i have the same order and the same size.

• F -rotation distance

  The F -rotation distance is a distance on the set G _F (n, m) of all graphs with n vertices and m edges, containing a subgraph isomorphic to a given graph F of order at least 2 defined, for all G ₁, G ₂ ∈ G _F (m, n), as the minimum number of F-rotations necessary for transforming G ₁ into G ₂.

  An F-rotation is one of the edge transformations, defined as follows: let $F^{^{{\prime}} }$ be a subgraph of a graph G, isomorphic to F, let u, v, w be three distinct vertices of the graph G such that $u\not\in V (F^{^{{\prime}} })$, $v,w \in V (F^{^{{\prime}} })$, uv ∈ E(G), and $\mathit{uw}\notin E(G)$; H can be obtained from G by the F-rotation of the edge uv into the position uw if H = G −uv + uw.

• Binary relation distance

  Let R be a nonreflexive binary relation between graphs, i.e., R ⊂ G ×G, and there exists G ∈ G such that $(G,G)\notin R$.

  The binary relation distance is a metric (which can take the value ∞) on the set G of all graphs defined, for any graphs G ₁ and G ₂, as the minimum number of R-transformations necessary for transforming G ₁ into G ₂. We say that a graph H can be obtained from a graph G by an R-transformation if (H, G) ∈ R.

  An example is the distance between two triangular embeddings of a complete graph (i.e., its cellular embeddings in a surface with only 3-gonal faces) defined as the minimal number t such that, up to replacing t faces, the embeddings are isomorphic.

• Crossing-free transformation metrics

  Given a subset S of $\mathbb{R}^{2}$, a noncrossing spanning tree of S is a tree whose vertices are points of S, and edges are pairwise noncrossing straight line segments.

  The crossing-free edge move metric (see [AAH00]) on the set T _S of all noncrossing spanning trees of a set S, is defined, for any T ₁, T ₂ ∈ T _S , as the minimum number of crossing-free edge moves needed to transform T ₁ into T ₂. Such move is an edge transformation which consists of adding some edge e in T ∈ T _S and removing some edge f from the induced cycle so that e and f do not cross.

  The crossing-free edge slide metric is a metric on the set T _S of all noncrossing spanning trees of a set S defined, for any T ₁, T ₂ ∈ T _S , as the minimum number of crossing-free edge slides necessary for transforming T ₁ into T ₂. Such slide is one of the edge transformations which consists of taking some edge e in T ∈ T _S and moving one of its endpoints along some edge adjacent to e in T, without introducing edge crossings and without sweeping across points in S (that gives a new edge f instead of e). The edge slide is a special kind of crossing-free edge move: the new tree is obtained by closing with f a cycle C of length 3 in T, and removing e from C, in such a way that f avoids the interior of the triangle C.

• Traveling salesman tours distances

  The Traveling Salesman problem is the problem of finding the shortest tour that visits a set of cities. We will consider only Traveling Salesman problem with undirected links. For an n-city traveling salesman problem, the space $\mathcal{T}_{n}$ of tours is the set of $\frac{(n-1)!} {2}$ cyclic permutations of the cities $1,2,\ldots,n$.

  The metric D on $\mathcal{T}_{n}$ is defined in terms of the difference in form: if tours $T,T^{^{{\prime}} } \in \mathcal{T}_{n}$ differ in m links, then $D(T,T^{^{{\prime}} }) = m$.

  A k-OPT transformation of a tour T is obtained by deleting k links from T, and reconnecting. A tour $T^{^{{\prime}} }$, obtained from T by a k-OPT transformation, is called a k-OPT of T. The distance d on the set $\mathcal{T}_{N}$ is defined in terms of the 2-OPT transformations: $d(T,T^{^{{\prime}} })$ is the minimal i, for which there exists a sequence of i 2-OPT transformations which transforms T to $T^{^{{\prime}} }$. In fact, $d(T,T^{^{{\prime}} }) \leq D(T,T^{^{{\prime}} })$ for any $T,T^{^{{\prime}} } \in \mathcal{T}_{N}$ (see, for example, [MaMo95]). Cf. arc routing problems.

• Orientation distance

  The orientation distance (Chartrand–Erwin–Raines–Zhang, 2001) between two orientations D and D ^′ of a finite graph is the minimum number of arcs of D whose directions must be reversed to produce an orientation isomorphic to D ^′.

• Subgraphs distances

  The standard distance on the set of all subgraphs of a connected graph G = (V, E) is defined by

  $$\displaystyle{\min \{d_{\mathrm{path}}(u,v): u \in V (F),v \in V (H)\}}$$

  for any subgraphs F, H of G. For any subgraphs F, H of a strongly connected digraph D = (V, E), the standard quasi-distance is defined by

  $$\displaystyle{\min \{d_{\mathit{dpath}}(u,v): u \in V (F),v \in V (H)\}.}$$

  Using standard operations (rotation, shift, etc.) on the edge-set of a graph, one gets corresponding distances between its edge-induced subgraphs of given size which are subcases of similar distances on the set of all graphs of a given size and order.

  The edge rotation distance on the set S ^k(G) of all edge-induced subgraphs with k edges in a connected graph G is defined as the minimum number of edge rotations required to transform F ∈ S ^k(G) into H ∈ S ^k(G). We say that H can be obtained from F by an edge rotation if there exist distinct vertices u, v, and w in G such that uv ∈ E(F), uw ∈ E(G)∖E(F), and H = F −uv + uw.

  The edge shift distance on the set S ^k(G) of all edge-induced subgraphs with k edges in a connected graph G is defined as the minimum number of edge shifts required to transform F ∈ S ^k(G) into H ∈ S ^k(G). We say that H can be obtained from F by an edge shift if there exist distinct vertices u, v and w in G such that uv, vw ∈ E(F), uw ∈ E(G)∖E(F), and H = F −uv + uw.

  The edge move distance on the set S ^k(G) of all edge-induced subgraphs with k edges of a graph G (not necessary connected) is defined as the minimum number of edge moves required to transform F ∈ S ^k(G) into H ∈ S ^k(G). We say that H can be obtained from F by an edge move if there exist (not necessarily distinct) vertices u, v, w, and x in G such that uv ∈ E(F), wx ∈ E(G)∖E(F), and H = F −uv + wx. The edge move distance is a metric on S ^k(G). If F and H have s edges in common, then it is equal to k − s.

  The edge jump distance (which in general can take the value ∞) on the set S ^k(G) of all edge-induced subgraphs with k edges of a graph G (not necessary connected) is defined as the minimum number of edge jumps required to transform F ∈ S ^k(G) into H ∈ S ^k(G). We say that H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ∈ E(F), wx ∈ E(G)∖E(F), and H = F −uv + wx.

4 Distances on Trees

Let T be a rooted tree, i.e., a tree with one of its vertices being chosen as the root. The depth of a vertex v, depth(v), is the number of edges on the path from v to the root. A vertex v is called a parent of a vertex u, v = par(u), if they are adjacent, and depth(u) = depth(v) + 1; in this case u is called a child of v. A leaf is a vertex without child. Two vertices are siblings if they have the same parent.

The in-degree of a vertex is the number of its children. T(v) is the subtree of T, rooted at a node v ∈ V (T). If w ∈ V (T(v)), then v is an ancestor of w, and w is a descendant of v; nca(u, v) is the nearest common ancestor of the vertices u and v.

T is called a labeled tree if a symbol from a fixed finite alphabet $\mathcal{A}$ is assigned to each node. T is called an ordered tree if a left-to-right order among siblings in T is given. On the set $\mathbb{T}_{\mathit{rlo}}$ of all rooted labeled ordered trees there are three editing operations:

• Relabel—change the label of a vertex v;

• Deletion—delete a nonrooted vertex v with parent $v^{^{{\prime}} }$, making the children of v become the children of $v^{^{{\prime}} }$; the children are inserted in the place of v as a subsequence in the left-to-right order of the children of $v^{^{{\prime}} }$;

• Insertion—the complement of deletion; insert a vertex v as a child of a $v^{^{{\prime}} }$ making v the parent of a consecutive subsequence of the children of $v^{^{{\prime}} }$.

For unordered trees above operations (and so, distances) are defined similarly, but the insert and delete operations work on a subset instead of a subsequence.

We assume that there is a cost function defined on each editing operation, and the cost of a sequence of editing operations is the sum of the costs of these operations.

The ordered edit distance mapping is a representation of the editing operations. Formally, the triple (M, T ₁, T ₂) is an ordered edit distance mapping from T ₁ to T ₂, $T_{1},T_{2} \in \mathbb{T}_{\mathit{rlo}}$, if M ⊂ V (T ₁) × V (T ₂) and, for any (v ₁, w ₁), (v ₂, w ₂) ∈ M, the following conditions hold: v ₁ = v ₂ if and only if w ₁ = w ₂ (one-to-one condition), v ₁ is an ancestor of v ₂ if and only if w ₁ is an ancestor of w ₂ (ancestor condition), v ₁ is to the left of v ₂ if and only if w ₁ is to the left of w ₂ (sibling condition).

We say that a vertex v in T ₁ and T ₂ is touched by a line in M if v occurs in some pair in M. Let N ₁ and N ₂ be the set of vertices in T ₁ and T ₂, respectively, not touched by any line in M. The cost of M is given by $\gamma (M) =\sum _{(v,w)\in M}\gamma (v \rightarrow w) +\sum _{v\in N_{1}}\gamma (v \rightarrow \lambda ) +\sum _{w\in N_{2}}\gamma (\lambda \rightarrow w)$, where γ(a → b) = γ(a, b) is the cost of an editing operation a → b which is a relabel if $a,b \in \mathcal{A}$, a deletion if b = λ, and an insertion if a = λ. Here $\lambda \not\in \mathcal{A}$ is a special blank symbol, and γ is a metric on the set $\mathcal{A}\cup \lambda$ (excepting the value γ(λ, λ)).

• Tree edit distance

  The tree edit distance (see [Tai79]) on the set $\mathbb{T}_{\mathit{rlo}}$ of all rooted labeled ordered trees is defined, for any $T_{1},T_{2} \in \mathbb{T}_{\mathit{rlo}}$, as the minimum cost of a sequence of editing operations (relabels, insertions, and deletions) turning T ₁ into T ₂.

  In terms of ordered edit distance mappings, it is equal to $\min _{(M,T_{1},T_{2})}\gamma (M)$, where the minimum is taken over all such mappings (M, T ₁, T ₂).

  The unit cost edit distance between T ₁ and T ₂ is the minimum number of three above editing operations turning T ₁ into T ₂, i.e., it is the tree edit distance with cost 1 of any operation.

• Selkow distance

  The Selkow distance (or top-down edit distance, degree-1 edit distance) is a distance on the set $\mathbb{T}_{\mathit{rlo}}$ of all rooted labeled ordered trees defined, for any $T_{1},T_{2} \in \mathbb{T}_{\mathit{rlo}}$, as the minimum cost of a sequence of editing operations (relabels, insertions, and deletions) turning T ₁ into T ₂ if insertions and deletions are restricted to leaves of the trees (see [Selk77]).

  The root of T ₁ must be mapped to the root of T ₂, and if a node v is to be deleted (inserted), then any subtree rooted at v is to be deleted (inserted).

  In terms of ordered edit distance mappings, it is equal to $\min _{(M,T_{1},T_{2})}\gamma (M)$, where the minimum is taken over all such mappings (M, T ₁, T ₂) such that (par(v), par(w)) ∈ M if (v, w) ∈ M, where neither v nor w is the root.

• Restricted edit distance

  The restricted edit distance is a distance on the set $\mathbb{T}_{\mathit{rlo}}$ of all rooted labeled ordered trees defined, for any $T_{1},T_{2} \in \mathbb{T}_{\mathit{rlo}}$, as the minimum cost of a sequence of editing operations (relabels, insertions, and deletions) turning T ₁ into T ₂ with the restriction that disjoint subtrees should be mapped to disjoint subtrees.

  In terms of ordered edit distance mappings, it is equal to $\min _{(M,T_{1},T_{2})}\gamma (M)$, where the minimum is taken over all such mappings (M, T ₁, T ₂) satisfying the following condition: for all (v ₁, w ₁), (v ₂, w ₂), (v ₃, w ₃) ∈ M, nca(v ₁, v ₂) is a proper ancestor of v ₃ if and only if nca(w ₁, w ₂) is a proper ancestor of w ₃.

  This distance is equivalent to the structure respecting edit distance which is defined by $\min _{(M,T_{1},T_{2})}\gamma (M)$. Here the minimum is taken over all ordered edit distance mappings (M, T ₁, T ₂), satisfying the following condition: for all (v ₁, w ₁), (v ₂, w ₂), (v ₃, w ₃) ∈ M, such that none of v ₁, v ₂, and v ₃ is an ancestor of the others, nca(v ₁, v ₂) = nca(v ₁, v ₃) if and only if nca(w ₁, w ₂) = nca(w ₁, w ₃).

  Cf. constrained edit distance in Chap. 11.

• Alignment distance

  The alignment distance (see [JWZ94]) is a distance on the set $\mathbb{T}_{\mathit{rlo}}$ of all rooted labeled ordered trees defined, for any $T_{1},T_{2} \in \mathbb{T}_{\mathit{rlo}}$, as the minimum cost of an alignment of T ₁ and T ₂. It corresponds to a restricted edit distance, where all insertions must be performed before any deletions.

  Thus, one inserts spaces, i.e., vertices labeled with a blank symbol λ, into T ₁ and T ₂ so that they become isomorphic when labels are ignored; the resulting trees are overlaid on top of each other giving the alignment $T_{\mathcal{A}}$ which is a tree, where each vertex is labeled by a pair of labels. The cost of $T_{\mathcal{A}}$ is the sum of the costs of all pairs of opposite labels in $T_{\mathcal{A}}$.

• Splitting-merging distance

  The splitting-merging distance (see [ChLu85]) is a distance on the set $\mathbb{T}_{\mathit{rlo}}$ of all rooted labeled ordered trees defined, for any $T_{1},T_{2} \in \mathbb{T}_{\mathit{rlo}}$, as the minimum number of vertex splittings and mergings needed to transform T ₁ into T ₂.

• Degree-2 distance

  The degree-2 distance is a metric on the set $\mathbb{T}_{l}$ of all labeled trees (labeled free trees), defined, for any $T_{1},T_{2} \in \mathbb{T}_{l}$, as the minimum number of editing operations (relabels, insertions, and deletions) turning T ₁ into T ₂ if any vertex to be inserted (deleted) has no more than two neighbors. This metric is a natural extension of the tree edit distance and the Selkow distance.

A phylogenetic X-tree is an unordered unrooted tree with the labeled leaf set X and no vertices of degree two. If every interior vertex has degree three, the tree is called binary. Let $\mathbb{T}(X)$ denote the set of all phylogenetic X-trees.

• Robinson–Foulds metric

  A cut A | B of X is a partition of X into two subsets A and B (see cut semimetric). Removing an edge e from a phylogenetic X-tree induces a cut of the leaf set X which is called the cut associated with e.

  The Robinson–Foulds metric (or Bourque metric, bipartition distance) is a metric on the set $\mathbb{T}(X)$, defined, for any phylogenetic X-trees $T_{1},T_{2} \in \mathbb{T}(X)$, by

  $$\displaystyle{\frac{1} {2}\vert \Sigma (T_{1})\bigtriangleup \Sigma (T_{2})\vert = \frac{1} {2}\vert \Sigma (T_{1})\setminus \Sigma (T_{2})\vert + \frac{1} {2}\vert \Sigma (T_{2})\setminus \Sigma (T_{1})\vert,}$$

  where $\Sigma (T)$ is the collection of all cuts of X associated with edges of T.

  The Robinson–Foulds weighted metric is a metric on the set $\mathbb{T}(X)$ of all phylogenetic X-trees defined by

  $$\displaystyle{\sum _{A\vert B\in \Sigma (T_{1})\cup \Sigma (T_{2})}\vert w_{1}(A\vert B) - w_{2}(A\vert B)\vert }$$

  for all $T_{1},T_{2} \in \mathbb{T}(X)$, where $w_{i} = (w(e))_{e\in E(T_{i})}$ is the collection of positive weights, associated with the edges of the X-tree T _i , $\Sigma (T_{i})$ is the collection of all cuts of X, associated with edges of T _i , and w _i (A | B) is the weight of the edge, corresponding to the cut A | B of X, i = 1, 2. Cf. more general cut norm metric in Chap. 12 and rectangle distance on weighted graphs.

• μ -metric

  Given a phylogenetic X-tree T with n leaves and a vertex v in it, let $\mu (v) = (\mu _{1}(v),\ldots,\mu _{n}(v))$, where μ _i (v) is the number of different paths from the vertex v to the i-th leaf. Let μ(T) denote the multiset on the vertex-set of T with μ(v) being the multiplicity of the vertex v.

  The μ -metric (Cardona–Roselló–Valiente, 2008) is a metric on the set $\mathbb{T}(X)$ of all phylogenetic X-trees defined, for all $T_{1},T_{2} \in \mathbb{T}(X)$, by

  $$\displaystyle{\frac{1} {2}\vert \mu (T_{1})\Delta \mu (T_{2})\vert,}$$

  where $\Delta$ denotes the symmetric difference of multisets.

  Cf. the metrics between multisets in Chap. 1 and the Dodge–Shiode WebX quasi-distance in Chap. 22.

• Nearest neighbor interchange metric

  The nearest neighbor interchange metric (or crossover metric ) on the set $\mathbb{T}(X)$ of all phylogenetic X-trees, is defined, for all $T_{1},T_{2} \in \mathbb{T}(X)$, as the minimum number of nearest neighbor interchanges required to transform T ₁ into T ₂.

  A nearest neighbor interchange consists of swapping two subtrees in a tree that are adjacent to the same internal edge; the remainder of the tree is unchanged.

• Subtree prune and regraft distance

  The subtree prune and regraft distance is a metric on the set $\mathbb{T}(X)$ of all phylogenetic X-trees defined, for all $T_{1},T_{2} \in \mathbb{T}(X)$, as the minimum number of subtree prune and regraft transformations required to transform T ₁ into T ₂.

  A subtree prune and regraft transformation proceeds in three steps: one selects and removes an edge uv of the tree, thereby dividing the tree into two subtrees T _u (containing u) and T _v (containing v); then one selects and subdivides an edge of T _v , giving a new vertex w; finally, one connects u and w by an edge, and removes all vertices of degree two.

• Tree bisection-reconnection metric

  The tree bisection-reconnection metric (or TBR-metric ) on the set $\mathbb{T}(X)$ of all phylogenetic X-trees is defined, for all $T_{1},T_{2} \in \mathbb{T}(X)$, as the minimum number of tree bisection and reconnection transformations required to transform T ₁ into T ₂.

  A tree bisection and reconnection transformation proceeds in three steps: one selects and removes an edge uv of the tree, thereby dividing the tree into two subtrees T _u (containing u) and T _v (containing v); then one selects and subdivides an edge of T _v , giving a new vertex w, and an edge of T _u , giving a new vertex z; finally one connects w and z by an edge, and removes all vertices of degree two.

• Quartet distance

  The quartet distance (see [EMM85]) is a distance of the set $\mathbb{T}_{b}(X)$ of all binary phylogenetic X-trees defined, for all $T_{1},T_{2} \in \mathbb{T}_{b}(X)$, as the number of mismatched quartets (from the total number (₄ ⁿ) possible quartets) for T ₁ and T ₂.

  This distance is based on the fact that, given four leaves {1, 2, 3, 4} of a tree, they can only be combined in a binary subtree in three ways: (12 | 34), (13 | 24), or (14 | 23): the notation (12 | 34) refers to the binary tree with the leaf set {1, 2, 3, 4} in which removing the inner edge yields the trees with the leaf sets {1, 2} and {3, 4}.

• Triples distance

  The triples distance (see [CPQ96]) is a distance of the set $\mathbb{T}_{b}(X)$ of all binary phylogenetic X-trees defined, for all $T_{1},T_{2} \in \mathbb{T}_{b}(X)$, as the number of triples (from the total number (₃ ⁿ) possible triples) that differ (for example, by which leaf is the outlier) for T ₁ and T ₂.

• Perfect matching distance

  The perfect matching distance is a distance on the set $\mathbb{T}_{br}(X)$ of all rooted binary phylogenetic X-trees with the set X of n labeled leaves defined, for any $T_{1},T_{2} \in \mathbb{T}_{br}(X)$, as the minimum number of interchanges necessary to bring the perfect matching of T ₁ to the perfect matching of T ₂.

  Given a set $A =\{ 1,\ldots,2k\}$ of 2k points, a perfect matching of A is a partition of A into k pairs. A rooted binary phylogenetic tree with n labeled leaves has a root and n − 2 internal vertices distinct from the root. It can be identified with a perfect matching on 2n − 2, different from the root, vertices by following construction: label the internal vertices with numbers $n + 1,\ldots,2n - 2$ by putting the smallest available label as the parent of the pair of labeled children of which one has the smallest label among pairs of labeled children; now a matching is formed by peeling off the children, or sibling pairs, two by two.

• Tree rotation distance

  The tree rotation distance is a distance on the set T _n of all rooted ordered binary trees with n interior vertices defined, for all T ₁, T ₂ ∈ T _n , as the minimum number of rotations, required to transform T ₁ into T ₂.

  Given interior edges uv, vv ^′, vv ^{′ ′} and uw of a binary tree, the rotation is replacing them by edges uv, uv ^{′ ′}, vv ^′ and vw.

  There is a bijection between edge flipping operations in triangulations of convex polygons with n + 2 vertices and rotations in binary trees with n interior vertices.

• Attributed tree metrics

  An attributed tree is a triple (V, E, α), where T = (V, E) is the underlying tree, and α is a function which assigns an attribute vector α(v) to every vertex v ∈ V. Given two attributed trees (V ₁, E ₁, α) and (V ₂, E ₂, β), consider the set of all subtree isomorphisms between them, i.e., the set of all isomorphisms f: H ₁ → H ₂, H ₁ ⊂ V ₁, H ₂ ⊂ V ₂, between their induced subtrees.

  Given a similarity s on the set of attributes, the similarity between isomorphic induced subtrees is defined as $W_{s}(f) =\sum _{v\in H_{1}}s(\alpha (v),\beta (f(v)))$. Let ϕ be the isomorphism with maximal similarity W _s (ϕ) = W(ϕ).

  The following four semimetrics on the set T _att of all attributed trees are used:

  $$\displaystyle{\max \{\vert V _{1}\vert,\vert V _{2}\vert \}- W(\phi ),\,\,\,\,\vert V _{1}\vert + \vert V _{2}\vert - 2W(\phi )\,\,\,\mbox{ and }}$$
  $$\displaystyle{1 - \frac{W(\phi )} {\max \{\vert V _{1}\vert,\vert V _{2}\vert \}},\,\,\,\,1 - \frac{W(\phi )} {\vert V _{1}\vert + \vert V _{2}\vert - W(\phi )}.}$$

  They become metrics on the set of equivalences classes of attributed trees: two such trees (V ₁, E ₁, α) and (V ₂, E ₂, β) are called equivalent if they are attribute-isomorphic, i.e., if there exists an isomorphism g: V ₁ → V ₂ between the trees such that, for any v ∈ V ₁, we have α(v) = β(g(v)). Then | V ₁ |  =  | V ₂ |  = W(g).

• Maximal agreement subtree distance

  The maximal agreement subtree distance (MAST) is a distance of the set T of all trees defined, for all T ₁, T ₂ ∈ T, as the minimum number of leaves removed to obtain a (greatest) agreement subtree.

  An agreement subtree (or common pruned tree) of two trees is an identical subtree that can be obtained from both trees by pruning leaves with the same label.