Algorithms and Hardness for Metric Dimension on Digraphs

Abstract

In the Metric Dimension problem, one asks for a minimum-size set R of vertices such that for any pair of vertices of the graph, there is a vertex from R whose two distances to the vertices of the pair are distinct. This problem has mainly been studied on undirected graphs and has gained a lot of attention in the recent years. We focus on directed graphs, and show how to solve the problem in linear-time on digraphs whose underlying undirected graph (ignoring multiple edges) is a tree. This (nontrivially) extends a previous algorithm for oriented trees. We then extend the method to unicyclic digraphs (understood as the digraphs whose underlying undirected multigraph has a unique cycle). We also give a fixed-parameter-tractable algorithm for digraphs when parameterized by the directed modular-width, extending a known result for undirected graphs. Finally, we show that Metric Dimension is NP-hard even on planar triangle-free acyclic digraphs of maximum degree 6.

Research funded by the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20–25) and by the ANR project GRALMECO (ANR-21-CE48-0004).

A. Hakanen—Research supported by the Jenny and Antti Wihuri Foundation and partially by Academy of Finland grant number 338797.

1 Introduction

The metric dimension of a (di)graph G is the smallest size of a set of vertices that distinguishes all vertices of G by their vectors of distances from the vertices of the set. This concept was introduced in the 1970s by Harary and Melter [14] and by Slater [30] independently. Due to its interesting nature and numerous applications (such as robot navigation [18], detection in sensor networks [30] or image processing [22], to name a few), it has enjoyed a lot of attention. It also has been studied in the more general setting of metric spaces [3], and is generally part of the rich area of identification problems of graphs and other discrete structures [20].

More formally, let us denote by ${{\,\textrm{dist}\,}}(x,y)$ the distance from x to y in a digraph. Here, the distance ${{\,\textrm{dist}\,}}(x,y)$ is taken as the length of a shortest directed path from x to y; if no such path exists, ${{\,\textrm{dist}\,}}(x,y)$ is infinite, and we say that y is not reachable from x. We say that a set S is a resolving set of a digraph G if for any pair of distinct vertices v, w from G, there is a vertex x in S with ${{\,\textrm{dist}\,}}(x,v)\ne {{\,\textrm{dist}\,}}(x,w)$. Furthermore, we require that every vertex of G is reachable from at least one vertex of S. The metric dimension of G is the smallest size of a resolving set of G, and a minimum-size resolving set of G is called a metric basis of G.¹

We denote by Metric Dimension the computational version of the problem: given a (di)graph G, determine its metric dimension.

For undirected graphs, Metric Dimension has been extensively studied, and its non-local nature makes it highly nontrivial from an algorithmic point of view. On the hardness side, Metric Dimension was shown to be NP-hard for planar graphs of bounded degree [7], split, bipartite and line graphs [9], unit disk graphs [17], interval and permutation graphs of diameter 2 [11], and graphs of pathwidth 24 [19]. On the positive side, it can easily be solved in linear time on trees [4, 14, 18, 30]. More involved polynomial-time algorithms exist for unicyclic graphs [29] and, more generally, graphs of bounded cyclomatic number [9]; outerplanar graphs [7]; cographs [9]; chain graphs [10]; cactus-block graphs [16]; bipartite distance-hereditary graphs [24]. There are fixed parameter tractable (FPT) algorithms for the undirected graph parameters max leaf number [8], tree-depth [13], modular-width [2] and distance to cluster [12], but FPT algorithms are highly unlikely to exist for parameters solution size [15] and feedback vertex set [12].

Due to the interest for Metric Dimension on undirected graphs, it is natural to ask what can be said in the context of digraphs. The metric dimension of digraphs was first studied in [5] under a somewhat restrictive definition; for our definitions, we follow the recent paper [1], in which the algorithmic aspects of Metric Dimension on digraphs have been addressed. We call oriented graph a digraph without directed 2-cycles. A directed acyclic digraph (DAG for short) has no directed cycles at all. The underlying multigraph of a digraph is the one obtained by ignoring the arc orientations; its underlying graph is obtained from it by ignoring multiple edges. In a digraph, a strongly connected component is a subgraph where every vertex is reachable from all other vertices. Note that for the Metric Dimension problem, undirected graphs can be seen as a special type of digraphs where each arc has a symmetric arc.

The NP-hardness of Metric Dimension was proven for oriented graphs in [27] and, more recently, for bipartite DAGs of maximum degree 8 and maximum distance 4 [1] (the maximum distance being the length of a longest directed path without shortcuts). A linear-time algorithm for Metric Dimension on oriented trees was given in [1].

Our Results. We generalize the linear-time algorithm for Metric Dimension on oriented trees from [1] to all digraphs whose underlying graph is a tree. In other words, here we allow 2-cycles. This makes a significant difference with oriented trees, and as a result our algorithm is nontrivial. We then extend the used methods to solve Metric Dimension in linear time for unicyclic digraphs (digraphs with a unique cycle). Then, we prove that Metric Dimension can be solved in time $f(t)n^{O(1)}$ for digraphs of order n and modular-width t (a parameter recently introduced for digraphs in [31]). This extends the same result for undirected graphs from [2], and is the first FPT algorithm for Metric Dimension on digraphs. Finally, we complement the hardness result from [1] by showing that Metric Dimension is NP-hard even for planar triangle-free DAGs of maximum degree 6 and maximum distance 4. For results marked with $(*)$, we omit the full proof due to space constraints; those can be found in [6].

2 Digraphs Whose Underlying Graph is a Tree

For the sake of convenience, we call di-tree a digraph whose underlying graph is a tree. Trees are often the first non-trivial class to study for a graph problem. Metric Dimension is no exception to this, having been studied in the first papers for the undirected [4, 14, 18, 30] and the oriented [1] cases. In the undirected case, a minimum-size resolving set can be found by taking, for each vertex of degree at least 3 spanning k legs, the endpoint of $k-1$ of its legs (a leg is an induced path spanning from a vertex of degree at least 3, having its inner vertices of degree 2, and ending in a leaf). In the case of oriented trees, taking all the sources (a source is a vertex with no in-neighbour) and $k-1$ vertices in each set of k in-twins yields a metric basis (two vertices are in-twins if they have the same in-neighbourhood). Our algorithm, being on di-trees (which include both undirected trees and oriented trees), will reuse those strategies, but we will need to refine them in order to obtain a metric basis. The first refinement is of the notion of in-twins:

Definition 1

A strongly connected component E of a di-tree is an escalator if it satisfies the following conditions:

• 1. its underlying graph is a path with vertices $e_1,\ldots ,e_k$ ($k \ge 2$);

• 2. there is a unique vertex $y \not \in E$ such that the arc $\overrightarrow{ye_1}$ (resp. $\overrightarrow{ye_k}$) exists;

• 3. there can be any number (possibly, zero) of vertices $z \not \in E$ such that the arc $\overrightarrow{e_kz}$ (resp. $\overrightarrow{e_1z}$) exists; for every $i \in \{1,\ldots ,k-1\}$ (resp. $i \in \{2,\ldots ,k\}$), no arc $\overrightarrow{e_iz}$ with $z \not \in E$ exists.

Definition 2

In a di-tree, a set of vertices $A=\{a_1,\ldots ,a_k\}$ is a set of almost-in-twins if there is a vertex x such that:

• 1. for every $i \in \{1,\ldots ,k\}$, the arc $\overrightarrow{xa_i}$ exists and the arc $\overrightarrow{a_ix}$ does not exist;

• 2. for every $i \in \{1,\ldots ,k\}$, either $a_i$ is a trivial strongly connected component and $N^-(a_i)=\{x\}$, or $a_i$ is the endpoint of an escalator and $N^-(a_i)=\{x,y\}$ where y is its neighbour in the escalator.

Note that regular in-twins are also almost-in-twins. The second refinement is the following (for a given vertex x in a strongly connected component with C as an underlying graph, we call $d_C(x)$ the degree of x in C):

Definition 3

Given the underlying graph C of a strongly connected component of a di-tree and a set D of vertices, we call a set S of vertices inducing a path of order at least 2 in C a special leg if it verifies the four following properties:

• 1. S has a unique vertex v such that $v \in D$ or $d_C(v) \ge 3$;

• 2. S has a unique vertex w such that $d_C(w) = 1$, furthermore $w \not \in D$: w is called the endpoint of S;

• 3. all of the other vertices x of S verify $d_C(x) = 2$ and $x \not \in D$;

• 4. at least one of the vertices $y \in S \setminus \{w\}$ has an out-arc $\overrightarrow{yz}$ with $z \notin C$.

Note that several special legs can span from the same vertex, from which regular legs can also span. Algorithm 1, illustrated in Fig. 1, computes a metric basis of a di-tree.

Explanation of Algorithm 1. The algorithm will compute a metric basis $\mathcal {B}$ of a di-tree T in linear-time. The first thing we do is to add every source in T to $\mathcal {B}$ (line 1). Then, for every set of almost-in-twins, we add all of them but one to $\mathcal {B}$ (lines 2–3). Those two first steps, depicted in Fig. 1a, are the ones used to compute the metric basis of an orientation of a tree [1], and as such they are still necessary for managing the non-strongly connected components of the di-tree. Note that we are specifically managing sets of almost-in-twins, which include sets of in-twins, since it is necessary to resolve the specific case of escalators. The rest of the algorithm consists in managing the strongly connected components.

For each strongly connected component having C as an underlying graph, we first identify each vertex x of C that has an in-arc coming from outside C. Indeed, since x is the “last” vertex of a path coming from outside C, there are vertices of $\mathcal {B}$ “behind” this in-arc (or they can themselves be a vertex in $\mathcal {B}$), which we will call $\mathcal {B}_x$. However, the vertices in $\mathcal {B}_x$ can be “projected” on x since, T being a di-tree, x is on every shortest path from the vertices of $\mathcal {B}$ “behind”the in-arc to the vertices of C. Hence, we will mark x as a dummy vertex (lines 5–7, depicted in Figure 1b): we will consider that it is in $\mathcal {B}$ for the rest of this step, and acts as a representative of the set $\mathcal {B}_x$ with respect to C.

We then have to manage some specific cases whenever C is a path (lines 8–17). Indeed, the last two steps of the algorithm do not always work under some conditions. Those specific conditions are highlighted in the proof.

The last two steps are then applied. First, we have to consider the special legs defined in Definition 3. The idea behind those special legs is the following: for every out-arc $\overrightarrow{yz}$ with y in the special leg and z outside of C, any vertex in the metric basis “before” the start of the special leg will not distinguish z and the next neighbour of y in the special leg. Hence, we have to add at least one vertex to $\mathcal {B}$ for each special leg, and we choose the endpoint of the special leg (lines 18–19, depicted in Fig. 1c). Finally, we apply the well-known algorithm for computing the metric basis of a tree to the remaining parts of C (lines 20–21, depicted in Fig. 1d). The special legs and the legs containing a dummy vertex, being already resolved, are not considered in this part.

Theorem 4

$(*)$. Algorithm 1 computes a metric basis of a di-tree in linear time.

Fig. 1.

Illustration of Algorithm 1. For the sake of simplicity, there are only two strongly connected components, for which we only represent the underlying graph with bolded edges, so every bolded edge is a 2-cycle. One of the two strongly connected components is a simple path that does not require any action. Vertices in the metric basis are colored in red. (Color figure online)

3 Orientations of Unicyclic Graphs

A unicyclic graph U is constituted of a cycle C with vertices $c_1,\ldots ,c_n$, and each vertex $c_i$ is the root of a tree $T_i$ (we can have $T_i$ be simply the isolated $c_i$ itself). The metric dimension of an undirected unicyclic graph has been studied in [26, 28, 29]. In [26], Poisson and Zhang proved bounds for the metric dimension of a unicyclic graph in terms of the metric dimension of a tree we obtain by removing one edge from the cycle. Sedlar and Škrekovski showed more recently that the metric dimension of a unicyclic graph is one of two values in [28], and then the exact value of the metric dimension based on the structure of the graph in [29]. In this section, we will show that one can compute a metric basis of an orientation of a unicyclic graph in linear time. The algorithm mostly consists in using sources and in-twins, with a few specific edge cases to consider.

In this section, an induced directed path $\overrightarrow{P}$ is the orientation of an induced path with only one source and one sink which are its two endpoints. It is said to be spanning from u if u is its source endpoint, and its length is its number of edges. We also need the following definition:

Definition 5

Let $\overrightarrow{U}$ be the orientation of a unicyclic graph. Given an orientation of a cycle $\overrightarrow{C}$ of even length $n=2k$ with two sources, if its sources are $c_i$ and $c_{i+2}$, its sinks are $c_{i+1}$ and $c_{i+1+k}$, and there are, in $\overrightarrow{C} \setminus \{c_i,c_{i+2}\}$, neither in-twins nor in-arcs coming from outside of $\overrightarrow{C}$, we call an induced directed path $\overrightarrow{P}$ an concerning path if it verifies the three following properties:

• 1. $\overrightarrow{P}$ spans from $c_{i+1}$;

• 2. $\overrightarrow{P}$ has length $k-2$;

• 3. $\overrightarrow{P}$ has no in-arc coming from outside of $\overrightarrow{P} \cup \overrightarrow{C}$;

Furthermore, if, for every vertex in $\overrightarrow{P}$ belonging to a nonempty set I of in-twins, every vertex in I belongs to a concerning path, then, we call $\overrightarrow{P}$ an unfixable path.

A path that is a concerning path, but not an unfixable path, will be called a fixable path.

Finally, a vertex might belong both to an unfixable path and to a fixable path; in this case, the fixable path takes precedence (i.e., we will consider that the vertex belongs to the fixable path).

Explanation of Algorithm 2. The algorithm will compute a metric basis $\mathcal {B}$ of an orientation $\overrightarrow{U}$ of a unicyclic graph U in linear-time. The first thing we do is to add every source in $\overrightarrow{U}$ to $\mathcal {B}$ (line 1). We will also manage the sets of in-twins in $\overrightarrow{U}$ (lines 3–7), which we need to do after taking care of some special cases that might influence the choice of in-twins. When we have the choice, we prioritize taking in-twins that are in the cycle to guarantee reachability of vertices in the cycle. Note that those two sets (along with the right priority) are enough in most cases.

We then have to manage six specific cases (line 2). Those special cases are handled in Algorithm 3. The first two special cases occur when the cycle has no sink. First, if the cycle has no sink, no in-twin, and no arc coming from outside, then, we have to add one vertex of the cycle to $\mathcal {B}$ in order to maintain reachability (lines 1–2). Then, if the cycle has no sink, only one in-arc $\overrightarrow{uc_i}$ is coming from outside of it, and there is a vertex v with $N^-(v)=\{u\}$, then, we have to add either $c_i$ or v to $\mathcal {B}$ in order to resolve them (lines 3–4).

The next three special cases occur when the cycle has one sink. First, if there is only one sink in the cycle, it is an out-neighbour of the source, and no vertex from the cycle apart from the source is an in-twin or has an in-arc coming from outside of the cycle, then we need to add one of the out-neighbours of the source in the cycle to $\mathcal {B}$ in order to resolve them (lines 6–7).

Then, there are two specific cases when the cycle has one sink, both based on the same principle. Both happen when the source is $c_i$, the sink is $c_{i+k}$, it has no in-arc, and the cycle contains at least 2k vertices. In the fourth special case (lines 8–9), the vertex $c_{i+k-1}$ has an out-neighbour v verifying $N^-(v)=\{c_{i+k-1}\}$. We can see that, if no vertex in the other path from $c_i$ to $c_{i+k}$ (the path going through $c_{i-1},c_{i-2},\ldots ,c_{i+k+1}$) is in $\mathcal {B}$, then, v and $c_{i+k}$ will not be resolved. Those vertices can be added to $\mathcal {B}$ if they have an in-arc or if they are an in-twin (they will have priority). However, note that $c_{i-1}$ might be an in-twin of $c_{i+1}$, in which case it should be added to $\mathcal {B}$, resolving the conflict. Hence, if none of $c_{i-1},c_{i-2},\ldots ,c_{i+k+1}$ has an in-arc or is an in-twin, then, we can add $c_{i-1}$ to $\mathcal {B}$ in order to resolve v and $c_{i+k}$. Note that, in this case, in comparison to just the sources and the resolution of sets of in-twins, we add one more vertex to $\mathcal {B}$ if $c_{i-1}$ is the only in-twin of $c_{i+1}$. The same reasoning can be made with the symmetric case.

The fifth special case (lines 10–11) occurs when the cycle contains exactly 2k vertices and both $c_{i+k-1}$ and $c_{i+k+1}$ have an out-neighbour (respectively $v_-$ and $v_+$) with in-degree 1: the pairs of vertices $(v_-,c_{i+k})$ and $(v_+,c_{i+k})$ might not be resolved. We can see that any in-arc or in-twin along a path from $c_i$ to $c_{i+k}$ will resolve $c_{i+k}$ and the v pendant on the other path (thus either fully resolving those two pairs, or bringing us back to the previous special case), except if $c_{i-1}$ and $c_{i+1}$ are the only in-twins in the cycle and if they do not have another in-twin. Hence, if no vertex from the cycle except $c_i$ has an in-arc, no vertex from the cycle except $c_i$, $c_{i-1}$ and $c_{i+1}$ is an in-twin, and $c_{i-1}$ and $c_{i+1}$ do not have another in-twin, then, we need to add at least one more vertex to $\mathcal {B}$ in order to resolve the two pairs of vertices, and adding $c_{i+k}$ does exactly that.

Finally, the sixth special case is more complex (lines 12–13 and consideration in the choice of in-twins). Assume that the cycle $\overrightarrow{C}$ is of even length n, has neither in-twin nor in-arc coming from outside (except the sources), and that there are two sinks in the $\overrightarrow{C}$: one at distance 1 from the sources, and the other at the opposite end of $\overrightarrow{C}$. Now, if the first sink has spanning concerning paths, then, the second cycle and the endpoints of those concerning paths might not be resolved, since they are at the same distance ($\frac{n}{2}-1$) of both sources of $\overrightarrow{C}$. Thus, we need to apply a strategy in order to resolve those vertices while trying to not add a supplementary vertex to $\mathcal {B}$. This is done by considering the two kinds of concerning paths, and having a priority in the selection of in-twins.

All the other cases of the cycle are already resolved through the sources and in-twins steps.

Theorem 6

$(*)$. Algorithm 2 computes a metric basis of an orientation of a unicyclic graph in linear time.

4 Modular Width

In a digraph G, a set $X \subseteq V(G)$ is a module if every vertex not in X ’sees’ all vertices of X in the same way. More precisely, for each $v \in V(G) \setminus X$ one of the following holds: (i) $(v,x),(x,v) \in E(G)$ for all $x \in X$, (ii) $(v,x),(x,v) \notin E(G)$ for all $x \in X$, (iii) $(v,x) \in E(G)$ and $(x,v) \notin E(G)$ for all $x \in X$, (iv) $(v,x) \notin E(G)$ and $(x,v) \in E(G)$ for all $x \in X$. The singleton sets, $\emptyset$, and V(G) are trivially modules of G. We call the singleton sets the trivial modules of G.

The graph G[X] where X is a module of G is called a factor of G. A family $\mathcal {X} = \{X_1, \ldots , X_s\}$ is a factorization of G if $\mathcal {X}$ is a partition of V(G), and each $X_i$ is a module of G. If X and Y are two non-intersecting modules, then the relationship between $x \in X$ and $y \in Y$ is one of (i)-(iv) and always the same no matter which vertices x and y are exactly. Thus, given a factorization $\mathcal {X}$, we can identify each module with a vertex, and connect them to each other according to the arcs between the modules. More formally, we define the quotient $G / \mathcal {X}$ with respect to the factorization $\mathcal {X}$ as the graph with the vertex set $\mathcal {X} = \{X_1, \ldots , X_s\}$ and $(X_i,X_j) \in E(G / \mathcal {X})$ if and only if $(x_i,x_j) \in E(G)$ where $x_i \in X_i$ and $x_j \in X_j$. A quotient depicts the connections of the different modules of a factorization to each other while omitting the internal structure of the factors. Each factor itself can be factorized further (as long as it is nontrivial, i.e. not a single vertex). By factorizing the graph G and its factors until no further factorization can be done, we obtain a modular decomposition of G. The width of a decomposition is the maximum number of sets in a factorization (or equivalently, the maximum number of vertices in a quotient) in the decomposition. The modular width of G is defined as the minimum width over all possible modular decompositions of G, and we denote it by $\textrm{mw}(G)$. An optimal modular decomposition of a digraph can be computed in linear time [21]. Metric Dimension for undirected graphs was shown to be fixed parameter tractable when parameterized by modular width by Belmonte et al. [2]. We will generalize their algorithm to directed graphs.

The following result lists several useful observations.

Proposition 7

$(*)$. Let $\mathcal {X} = \{X_1, \ldots , X_s\}$ be a factorization of G, and let $W \subseteq V(G)$ be a resolving set of G.

• (i) For all $x,y \in X_i$ and $z \in X_j$, $i\ne j$, we have ${{\,\textrm{dist}\,}}_G(x,z) = {{\,\textrm{dist}\,}}_G(y,z)$ and ${{\,\textrm{dist}\,}}_G(z,x) = {{\,\textrm{dist}\,}}_G(z,y)$.

• (ii) For all $x \in X_i$ and $y \in X_j$, $i\ne j$, we have ${{\,\textrm{dist}\,}}_G(x,y) = {{\,\textrm{dist}\,}}_{G/\mathcal {X}}(X_i,X_j)$.

• (iii) For all $x,y \in V(G)$ we have either ${{\,\textrm{dist}\,}}_G(x,y) \le \textrm{mw}(G)$ or ${{\,\textrm{dist}\,}}_G(x,y) = \infty$.

• (iv) The set $\{X_i \in \mathcal {X} \, | \, W \cap X_i \ne \emptyset \}$ is a resolving set of the quotient $G / \mathcal {X}$.

• (v) For all distinct $x,y \in X_i$, where $X_i \in \mathcal {X}$ is nontrivial, we have ${{\,\textrm{dist}\,}}_G(w,x) \ne {{\,\textrm{dist}\,}}_G(w,y)$ for some $w \in W \cap X_i$.

• (vi) Let $w_1,w_2 \in X_i$. If ${{\,\textrm{dist}\,}}_G(w_1,x) \ne {{\,\textrm{dist}\,}}_G(w_2,x)$, then $x \in X_i$ and ${{\,\textrm{dist}\,}}_G(w_1,x) \ne {{\,\textrm{dist}\,}}_G(w_1,y)$ or ${{\,\textrm{dist}\,}}_G(w_2,x) \ne {{\,\textrm{dist}\,}}_G(w_2,y)$ for each $y \notin X_i$.

The basic idea of our algorithm (and that of [2]) is to compute metric bases that satisfy certain conditions for the factors and combine these local solutions into a global solution. We know that nontrivial modules must contain elements of a resolving set, as modules must be resolved locally (Proposition 7 (i)). While combining the local solutions of nontrivial modules, we need to make sure that a vertex $x \in X_i$, where $X_i$ is nontrivial, is resolved from all $y \notin X_i$. If x and y are resolved as described in Proposition 7 (vi), then we need to do nothing special. However, if $x \in X_i$ is such that ${{\,\textrm{dist}\,}}_G(w,x) = d$ for all $w \in W_i$ and a fixed $d \in \{1, \ldots , \textrm{mw}(G), \infty \}$, there might exist a vertex $y \notin X_i$ such that $W_i$ does not resolve x and y. We call such a vertex x d-constant (with respect to $W_i$). We need to keep track of d-constant vertices and make sure they are resolved when we combine the local solutions. There are at most $\textrm{mw}(G) + 1$ d-constant vertices in each factor due to Proposition 7 (iii). We need to also make sure vertices in different modules that contain no elements of the solution set are resolved. To do this, we might need to include some vertices from the trivial modules in addition to the vertices we have included from the nontrivial modules.

In the algorithm presented in [2], the problems described above are dealt with by computing values w(H, p, q) for every factor H, where w(H, p, q) is the minimum cardinality of a resolving set of H (with respect to the distance in G) where some vertex is 1-constant iff $p = true$ and some vertex is 2-constant iff $q = true$ (for undirected graphs these are the only two relevant cases). The same values are then computed for the larger graph by combining different solutions of the factors and taking their minimum. Our generalization of this algorithm is along the same lines as the original, however, we have more boolean values to keep track of. One difference to the techniques of the original algorithm is that we do not use the auxiliary graphs Belmonte et al. use. These auxiliary graphs were needed to simulate the distances of the vertices of a factor in G as opposed to only within the factor. In our approach, we simply use the distances in G.

Theorem 8

$(*)$. The metric dimension of a digraph G with $\textrm{mw}(G) \le t$ can be computed in time $\mathcal {O}(t^5 2^{t^2} n + n^3 + m )$ where $n=|V(G)|$ and $m = |E(G)|$.

Proof

(sketch). Let us consider one level of an optimal modular decomposition of G. Let H be a factor somewhere in the decomposition, and let $\mathcal {X} = \{X_1, \ldots , X_s\}$ be the factorization of H according to the modular decomposition. For the graph H (and its nontrivial factors $H[X_i]$) we denote by $w(H,\textbf{p})$ the minimum cardinality of a set $W \subseteq V(H)$ such that

• (i) W resolves V(H) in G,

• (ii) $\textbf{p}= (p_1,\ldots ,p_{\textrm{mw}(G)},p_\infty )$ where $p_d = true$ if and only if H contains a d-constant vertex with respect to W.

If such a set does not exist, then $w(H,\textbf{p})=\infty$. In order to compute the values $w(H,\textbf{p})$, we next introduce the auxiliary values $\omega (\textbf{p},I,P)$. The values $w(H[X_i],\textbf{p})$ are assumed to be known for all $\textbf{p}$ and nontrivial modules $X_i$. Let the factorization $\mathcal {X}$ be labeled so that the modules $X_i$ are trivial for $i \in \{1, \ldots , h\}$ and nontrivial for $i \in \{h+1, \ldots , s\}$. Let $I \subseteq \{1, \ldots , h\}$ and $P = \left( \textbf{p}^{h+1}, \ldots \textbf{p}^{s} \right) ^T$.

We define $\omega (\textbf{p},I,P) = |I| + \sum _{i=h+1}^{s} w(H[X_i],\textbf{p}^i)$ if the conditions (a)–(d) hold. In what follows, a representative of a module $X_i$ is denoted by $x_i$.

• (a) The set $Z=\{X_i \in \mathcal {X} \, | \, i \in I \cup \{h+1,\ldots ,s\} \}$ resolves the quotient $H/\mathcal {X}$ with respect to the distances in G.

• (b) For $d \in \{1,\ldots ,\textrm{mw}(G),\infty \}$ and $i \in \{h+1,\ldots ,s\}$, if $p_d^i = true$, then for each trivial module $X_j = \{x_j\}$ where $j \notin I$ we have ${{\,\textrm{dist}\,}}_G(x_i,x_j) \ne d$ or there exists $X_k \in Z \setminus \{X_i\}$ such that ${{\,\textrm{dist}\,}}_G(x_k,x_i) \ne {{\,\textrm{dist}\,}}_G(x_k,x_j)$.

• (c) For $d_1,d_2 \in \{1,\ldots ,\textrm{mw}(G),\infty \}$ and distinct $i,j \in \{h+1,\ldots ,s\}$, if $p_{d_1}^i = p_{d_2}^j = true$, then ${{\,\textrm{dist}\,}}_G(x_i,x_j) \ne d_1$, or ${{\,\textrm{dist}\,}}_G(x_j,x_i) \ne d_2$, or there exists $X_k \in Z \setminus \{X_i, X_j\}$ such that ${{\,\textrm{dist}\,}}_G(x_k,x_i) \ne {{\,\textrm{dist}\,}}_G(x_k,x_j)$.

• (d) For all $d \in \{1,\ldots ,\textrm{mw}(G),\infty \}$, we have $p_d = true$ (in $\textbf{p}$) if and only if for some $i \in \{1, \ldots , h\} \setminus I$ we have ${{\,\textrm{dist}\,}}_G(x_j,x_i) = d$ for all $X_j \in Z$, or for some $i \in \{h+1, \ldots , s\}$ we have $p_d^i= true$ and ${{\,\textrm{dist}\,}}_G(x_j,x_i) = d$ for all $X_j \in Z \setminus \{X_i\}$.

If these conditions cannot be met, then we set $\omega (\textbf{p},I,P) = \infty$.

When we know all the values $\omega (\textbf{p},I,P)$, we can easily calculate $w(H,\textbf{p})$ since $w(H,\textbf{p}) = \min _{I,P} \omega (\textbf{p},I,P)$ (proof omitted due to lack of space).

To conclude the proof, we note that $w(G,\textbf{p})$ is the minimum cardinality of a resolving set that gives some vertices the specific distance combinations according to $\textbf{p}$. Thus, the metric dimension of G is $\min w(G,\textbf{p})$ where the minimum is taken over all $\textbf{p}$ such that $p_\infty = false$.    $\square$

5 NP-Hardness for Restricted DAGs

We now complement the hardness result from [1], which was for bipartite DAGs of maximum degree 8 and maximum distance 4.

Theorem 9

$(*)$. Metric Dimension is NP-complete, even on planar triangle-free DAGs of maximum degree 6 and maximum distance 4.

Proof

(sketch). We reduce from Vertex Cover on 2-connected planar cubic graphs, which is known to be NP-complete [23, Theorem 4.1].

Given a 2-connected planar cubic graph G, we construct a DAG $G'$ as follows. First of all, note that by Petersen’s theorem, G contains a perfect matching $M\subset E(G)$, that can be constructed in polynomial time. A planar embedding of G can also be constructed in polynomial time, so we fix one. We let $V(G')=V(G)\bigcup _{e=uv\in E(G)}\{a_e,b_e,c_e,d_e^u,d_e^v\}\bigcup _{e=uv\in M}\{f_e,g_e,h_e\}$. For every edge $e=uv$ of G, we add the arcs $\{\overrightarrow{a_eb_e},\overrightarrow{b_ec_e},\overrightarrow{c_ed_e^u},\overrightarrow{c_ed_e^v},\overrightarrow{ud_e^u},\overrightarrow{vd_e^v}\}$. For every edge $e=uv$ of the perfect matching M of G, assuming the neighbours of u (in the clockwise cyclic order with respect to the planar embedding of G) are v, x, y and those of v are u, s, t, we arbitrarily fix one side of the edge uv to place the vertices $f_e$, $g_e$ and $h_e$ (say, on the side that is close to the edges ux and vt). We add the arcs $\{\overrightarrow{f_eg_e},\overrightarrow{g_ec_e},\overrightarrow{g_eh_e},\overrightarrow{h_eu},\overrightarrow{h_ev},\overrightarrow{c_ec_{uy}},\overrightarrow{c_ec_{vs}},\overrightarrow{h_ec_{ux}},\overrightarrow{h_ec_{vt}}\}$.

Using the embedding of G, $G'$ can also be drawn in a planar way, it has maximum degree 6 (the vertices of type $c_e$ are of degree 6), has no triangles, and no shortest directed path of length 4.

We claim that G has a vertex cover of size at most k if and only if $G'$ has metric dimension at most $k+|E(G)|+|M|$ (proof omitted due to lack of space).

   $\square$

6 Conclusion

Metric Dimension can be solved in polynomial time on outerplanar graphs, using an involved algorithm [7]. Can one generalize our algorithms for trees and unicyclic graphs to solve Metric Dimension for directed (or at least, oriented) outerplanar graphs in polynomial time? Extending our algorithm to cactus graphs already seems nontrivial.

Is Metric Dimension NP-hard on planar bipartite subcubic DAGs?

Also, it would be interesting to see which hardness results known for Metric Dimension of undirected graphs also hold for DAGs, or for oriented graphs.