On the Distance Identifying Set Meta-problem and Applications to the Complexity of Identifying Problems on Graphs

Abstract

Numerous problems consisting in identifying vertices in graphs using distances are useful in domains such as network verification and graph isomorphism. Unifying them into a meta-problem may be of main interest. We introduce here a promising solution named Distance Identifying Set. The model contains Identifying Code (IC), Locating Dominating Set (LD) and their generalizations r-IC and r-LD where the closed neighborhood is considered up to distance r. It also contains Metric Dimension (MD) and its refinement r-MD in which the distance between two vertices is considered as infinite if the real distance exceeds r. Note that while IC = 1-IC and LD = 1-LD, we have MD = $\infty$-MD; we say that MD is not local. In this article, we prove computational lower bounds for several problems included in Distance Identifying Set by providing generic reductions from (Planar) Hitting Set to the meta-problem. We focus on two families of problems from the meta-problem: the first one, called local, contains r-IC, r-LD and r-MD for each positive integer r while the second one, called 1-layered, contains LD, MD and r-MD for each positive integer r. We have: (1) the 1-layered problems are NP-hard even in bipartite apex graphs, (2) the local problems are NP-hard even in bipartite planar graphs, (3) assuming ETH, all these problems cannot be solved in $2^{o(\sqrt{n})}$ when restricted to bipartite planar or apex graph, respectively, and they cannot be solved in $2^{o(n)}$ on bipartite graphs, and (4) except if ${\mathsf{W}[0]} = {\mathsf{W}[2]}$, they do not admit parameterized algorithms in $2^{{\mathcal {O}}(k)} \cdot n^{{\mathcal {O}}(1)}$ even when restricted to bipartite graphs. Here k is the solution size of a relevant identifying set. In particular, Metric Dimension cannot be solved in $2^{o(n)}$ under ETH, answering a question of Hartung and Nichterlein (Proceedings of the 28th conference on computational complexity, CCC, 2013).

1 Introduction and Corresponding Works

Problems consisting in identifying each element of a combinatorial structure with a hopefully small number of elements have been widely investigated. Here, we study a meta identification problem which generalizes three of the most well-known identification problems in graphs, namely Identifying Code (IC), Locating Dominating Set (LD) and Metric Dimension (MD). These problems are used in network verification [3, 4], fault-detection in networks [24, 30], graph isomorphism [2] or logical definability of graphs [25]. The versions of these problems in hypergraphs have been studied under different names in [6,7,8].

Given a graph G with vertex set V, the classical identifying sets are defined as follows:

• IC: Introduced by Karposky et al. [24], a set C of vertices of G is said to be an identifying code if none of the sets $N[v] \cap C$ are empty, for $v \in V$ and they are all distinct.

• LD: Introduced by Slater [27, 28], a set C of vertices of G is said to be a locating-dominating set if none of the sets $N[v] \cap C$ are empty, for $v \in V\setminus C$ and they are all distinct. When not considering the dominating property ($N[v] \cap C$ may be empty), these sets have been studied in [2] as distinguing sets and in [25] as sieves.

• MD: Introduced independently by Harary et al. [18] and Slater [26], a set C of vertices of G is said to be a resolving set if C contains one vertex from each connected component of G and, for every distinct vertices u and v of G, there exists a vertex w of C such that $d(w,u) \ne d(w,v)$. The metric dimension of G is the minimum size of its resolving sets.

The corresponding minimization problems of the previous identifying sets are defined as follows: given a graph G, compute a suitable set C of minimal size, if one exists. In this paper, we mainly focus on the computational complexity of these minimization problems.

Known results A wide collection of NP-hardness results has been proven for the problems.

For IC and LD, the minimization problems are indeed NP-hard [10, 11]. Charon et al. showed the NP-hardness when restricted to bipartite graphs [9], while Auger showed it for planar graphs with arbitrarily large girth [1]. For trees, there exists a linear algorithm [27].

Metric Dimension is also NP-hard, even when restricted to Gabriel unit disk graphs [17, 21]. Epstein et al. [14] showed that MD is polynomial on several classes as trees, cycles, cographs, partial wheels, and graphs of bounded cyclomatic number, but it remains NP-hard on split graphs, bipartite graphs, co-bipartite and line graphs of bipartite graphs. Diaz et al. [12] proved that MD is polynomial on outerplanar graphs whereas it remains NP-hard on bounded degree planar graphs. Additionally, Fernau et al. provided a linear algorithm for chain graphs [16] while Hoffmann et al. gave one for cactus block graphs [22]. Foucaud et al. [15] also proved the NP-hardness of the three problems restricted to interval graphs and permutation graphs.

These notions may be considered under the parameterized point of view; see [13] for a comprehensive study of Fixed Parameter Tractability (FPT). In the following, the parameter k is chosen as the solution size of a suitable set.

For IC and LD, the parameterized problems are clearly FPT since the number of vertices of a positive instance is bounded by $2^k+k$ (k vertices may characterize $2^k$ neighbors).

Such complexity is not likely to be achievable in the case of MD, since it would imply ${\mathsf{W}[2]} = \mathsf{FPT}~(= {\mathsf{W}[0]})$. Indeed, Hartung et al. [19, 20] showed MD is ${\mathsf{W}[2]}$-hard for bipartite subcubic graphs. The problem is however FPT on families of graphs with degree $\varDelta$ growing with the number of vertices because the size k of a resolving set must satisfy $log_3(\varDelta ) < k$. Foucaud et al. also provided a FPT algorithm on interval graphs [15], which was generalized to graphs of bounded tree-length by Belmonte et al. [5] (Fig. 1).

Fig. 1

The computational lower bounds implied by our generic reductions

Our contributions In order to unify the previous minimization problems, we introduce the concept of distance identifying functions. Given a distance identifying function f and a value r as a positive integer or infinity, the Distance Identifying Set meta-problem consists in finding a minimal sized r-dominating set which distinguishes every couple of vertices of an input graph thanks to the function f. Here, we mainly focus on two natural subfamilies of problems of Distance Identifying Set named local, in which a vertex cannot discern the vertices outside of its i-neighborhood, for i a fixed positive integer, and 1-layered, where a vertex is able to separate its open neighborhood from the distant vertices.

With this approach, we obtain several computational lower bounds for problems included in Distance Identifying Set by providing generic reductions from (Planar) Hitting Set to the meta-problem. The reductions rely on the classical set/element-gadget technique, the noteworthy adaptation of the clause/variable-gadget technique from SAT to Hitting Set.

We provide a 1-layered generic gadget as well as a local generic gadget. However, the local planar reduction is slightly more efficient than its 1-layered counterpart: it indeed implies computational lower bounds for planar graphs whereas the 1-layered reduction requires an auxiliary apex, limiting the consequences to apex graphs (see definition in Sect. 2.1).

The reductions in general graphs are designed to exploit the ${\mathsf{W}[2]}$-hardness of Hitting Set parameterized by the solution size $k_{HS}$ of a hitting set, hereby using:

Theorem 1

(folklore) Let$n_{HS}$and$m_{HS}$be the number of elements and sets of anHitting Setinstance, and$k_{HS}$be its solution size. A parameterized problem with parameterkadmitting a reduction fromHitting Setverifying$k = {\mathcal {O}}(k_{HS} + log(n_{HS} + m_{HS}))$does not have a parameterized algorithm running in$2^{{\mathcal {O}}(k)} \cdot n^{{\mathcal {O}}(1)}$time except if${\mathsf{W}[2]} = \mathsf{FPT}{}$.

Proof

Given a reduction from Hitting Set to a parameterized problem $\varPi$ such that the reduced parameter satisfies $k = {\mathcal {O}}(k_{HS} + log(n_{HS} + m_{HS}))$ and the size of the reduced instance verifies $n = (n_{HS} + m_{HS})^{{\mathcal {O}}(1)}$, an algorithm for $\varPi$ of running time $2^{{\mathcal {O}}(k)} \cdot n^{{\mathcal {O}}(1)}$ is actually an algorithm for Hitting Set of running time $2^{{\mathcal {O}}(k_{HS})} \cdot (n_{HS} + m_{HS})^{{\mathcal {O}}(1)}$, meaning that Hitting Set is FPT, a contradiction to its ${\mathsf{W}[2]}$-hardness (otherwise ${\mathsf{W}[2]} = \mathsf{FPT}$). $\square$

Hence, as each gadget contributes to the resulting solution size of a distance identifying set, we set up a binary compression of the gadgets to limit their number to the logarithm order. To the best of our knowledge, this merging gadgets technique has never been used to lower bound the parameterized complexity of a problem within the framework of W-hierarchy.

The organization of the paper is as follows. After a short reminder of the computational properties of Hitting Set, Sect. 2 contains the definitions of distance identifying functions and sets, allowing us to precise the computational lower bounds we obtain. The Sect. 3 designs the supports of the reductions as distance identifying graphs and compressed graph. Finally, the gadgets needed for the reductions to apply are given in Sect. 4 as well as the proofs of the main theorems.

2 Definition of the Meta-problem and Related Concepts

2.1 Preliminaries

Notations Throughout the paper, we consider simple non oriented graphs.

Given a positive integer n, the set of positive integers smaller than n is denoted by $[\![n]\!]$. By extension, we define $[\![\infty ]\!]= {\mathbb {N}}_{>0} \cup \{\infty \}$. Given two vertices u, v of a graph G, the distance between u and v corresponds to the number of vertices in the shortest path between u and v and is denoted d(u, v). The open neighborhood ofu is denoted by N(u), its closed neighborhood is $N[u] = N(u) \cup \{u\}$, and for a value $r \in [\![\infty ]\!]$, the r-neighborhood of u is $N_r[u]$, that is the set of vertices at distance less than $r+1$ of u. For $r = \infty$, the $\infty$-neighborhood of u is the set of vertices in the same connected component than u. We recall that a subset D of V is called an r-dominating set of G if for all vertices u of V, the set $N_r[u] \cap D$ is non-empty. Thus an $\infty$-dominating set of G contains at least a vertex for each connected component of G.

Given two subsets X and Y of V, the distance d(X, Y) corresponds to the value $d(X,Y) = \min \{d(x,y) ~ | ~ x \in X, ~y \in Y\}$. For a vertex u, we will also use d(u, X) and d(X, u), defined similarly. The symmetric difference between X and Y is denoted by $X \, \varDelta \, Y$, and the 2-combination of a set X is denoted ${\mathcal {P}}_2(X)$

Given two graphs $G = (V_G,E_G)$ and $H = (V_H,E_H)$, H is an induced subgraph ofG if $V_H \subseteq V_G$ and for all vertices u and v of $V_H$, $(u,v) \in E_G$ if and only if $(u,v) \in E_H$. We denote $H = G[V_H]$ and $V_G \setminus V_H$ by $V_{G \setminus H}$. Symmetrically, G is an induced supergraph ofH. A graph $G = (V \cup \{a\},E)$ is an apex graph if G[V] is planar, the vertex a being its apex.

The(Planar)Hitting Setproblem. Consider a universe of n elements denoted $\varOmega = \{u_i \; | \; i \in [\![n]\!]\}$ and a set of m non-empty subsets of $\varOmega$ denoted ${\mathcal {S}} = \{S_i \; | \; i \in [\![m]\!]\}$ such that every element belongs to at least a subset. Then, a subset of $\varOmega$ intersecting every set of ${\mathcal {S}}$ is called an hitting set of ${\mathcal {S}}$:

Hitting Set

Input: A universe $\varOmega$ and a set ${\mathcal {S}}$ of non-empty subsets of $\varOmega$ whose union covers $\varOmega$.

Output: A minimal-sized hitting set C of ${\mathcal {S}}$, i.e. a subset of $\varOmega$ satisfying $\forall S_i \in {\mathcal {S}}$, $S_i \cap C \ne \emptyset$.

The parameterized version Hitting Set(k) decides if there exists a hitting set of size k.

Theorem 2

(Downey and Fellows [13]) Hitting Setcannot be solved in$2^{o(n)}$time under ETH even if$m = {\mathcal {O}}(n)$. Moreover,Hitting Set(k) is ${\mathsf{W}[2]}$-hard.

Hitting Set may be translated into a dominating problem on bipartite graphs. Given an instance $(\varOmega , {\mathcal {S}})$ of Hitting Set, let us define $\phi (\varOmega , {\mathcal {S}}) = (V_{\varOmega } \cup V_{{\mathcal {S}}},E)$ as the bipartite graph of size $n+m$ such that for each $i \in [\![n]\!]$, there exists a vertex $v^{\varOmega }_i$ in $V_{\varOmega }$, for each $j \in [\![m]\!]$, there exists a vertex $v^{{\mathcal {S}}}_j$ in $V_{{\mathcal {S}}}$, and the edge $(v^{\varOmega }_i,v^{{\mathcal {S}}}_j)$ is present in E if and only if the element $u_i$ belongs to the subset $S_j$. Therefore, a hitting set of ${\mathcal {S}}$ is equivalent to a subset C of $V_{\varOmega }$ that dominates $V_{{\mathcal {S}}}$. We call $\phi (\varOmega , {\mathcal {S}})$the associated graph of$(\varOmega , {\mathcal {S}})$.

Planar Hitting Set

Input: An instance $(\varOmega , {\mathcal {S}})$ of Hitting Set such that $\phi (\varOmega , {\mathcal {S}})$ is planar.

Output: A hitting set C of ${\mathcal {S}}$ of minimal size.

We also consider the parameterized version PlanarHitting Set(k) of the latter problem.

Theorem 3

(folklore) There exists a reduction from the problemSATtoPlanarHitting Set(n) producing associated graphs of quadratic size in the numbernof variables of the instances ofSAT. ThusPlanarHitting Setcannot be solved in$2^{o(\sqrt{n})}$under ETH even if$m = {\mathcal {O}}(n)$.

Proof

Let $\varPhi$ be the set of the n variables present in the set ${\mathcal {C}}$ of clauses of an instance of SAT. For each variable $\varphi$ of $\varPhi$, we add two new elements $u_{\varphi }$ and ${\bar{u}}_{\varphi }$ to the universe $\varOmega _{\varPhi }$ representing the two possible affectations of variable $\varphi$, and we create a set $S_{\varphi } = \{u_{\varphi }, {\bar{u}}_{\varphi }\}$ that we append to the set ${\mathcal {S}}_{{\mathcal {C}}}$ of subsets of $\varOmega _{\varPhi }$. The independence of the sets $S_{\varphi }$ implies that the existence of a hitting set of size strictly smaller than n is impossible. Reciprocally, a potential hitting set of size exactly n must define an affectation of the n variables of $\varPhi$. Finally, to determine if an affectation satisfies the set of clauses ${\mathcal {C}}$, for each clause $c \in {\mathcal {C}}$ we append to ${\mathcal {S}}_{{\mathcal {C}}}$ the set of elements representing each literal present in the clause c. The equivalence between the satisfiability of ${\mathcal {C}}$ and the existence of a hitting set of ${\mathcal {S}}_{{\mathcal {C}}}$ of size n is immediate by construction. It remains to guarantee the planarity of the associated graph $\phi (\varOmega _{\varPhi },{\mathcal {S}}_{{\mathcal {C}}})$ . To do so, we actually apply the reduction on a restriction of SAT named Separate Simple Planar SAT (See [29] for a precise definition). Adding the sparsification lemma from [23], the reduction produces a graph of size linear in n, preserving the computational lower bound of Separate Simple Planar SAT. In particular, the latter problem is not solvable in $2^{o(\sqrt{n})}$ under ETH, $\square$

2.2 The Distance Identifying Set Meta-problem

Given a graph $G = (V,E)$ and $r \in [\![\infty ]\!]$, the classical identifying sets may be rewritten:

• r-IC: a subset C of V is an r-identifying code of G if it is an r-dominating set and for every distinct vertices u, v of V, a vertex w in C verifies $w \in N_r[u] \, \varDelta \, N_r[v]$.

• r-LD: a subset C of V is an r-locating dominating set of G if it is an r-dominating set and for every distinct vertices u, v of V, a vertex w in C verifies $w \in (N_r[u] \, \varDelta \, N_r[v]) \cup \{u,v\}$.

• r-MD: a subset C of V is an r-resolving set of G if it is an r-dominating set and for every distinct vertices u, v of V, a vertex w in C verifies $w \in N_r[u] \cup N_r[v]$ and $d(u,w) \ne d(v,w)$.

It is not hard to check that IC = 1-IC, LD = 1-LD (= 1-MD) and MD = $\infty$-MD.

A pattern clearly appears: the previous identifying sets only deviate on the criterion that the vertex w must verify. The pivotal idea is to consider an abstract version of the criterion which does not depend on the input graph. Hence:

Definition 1

(identifying function) A function f of type: $G \rightarrow (V \times {\mathcal {P}}_2(V) \rightarrow \{ {{\,\mathrm{true}\,}}, {{\,\mathrm{false}\,}}\})$, is called an identifying function. Given three vertices u, v and w of a graph G such that $u \ne v$, we write $f_G[w](u,v)$ to get the resulting boolean. The notation ${\mathcal {P}}_2(V)$ implies that $f_G$ is symmetric, that is $f_G[w](u,v) = f_G[w](v,u)$.

In the following, given an identifying function f and three vertices u, v, w of a graph G, we say that wf-distinguishes u and v if and only if $f_G[w](u,v)$ is ${{\,\mathrm{true}\,}}$. By extension, given three vertex sets C, X and Y of G, we say that Cf-distinguishes X and Y if for every u in X and v in Y, either $u = v$ or there exists w in C verifying $f_G[w](u,v)$. Finally, a graph G of vertex set V is f-distinguished by C when Cf-distinguishes V and V.

An identifying function may be defined using various criteria such as adjacency, coloring or vertex degree. Here, we focus on distance in order to mimic the proposed identifying sets.

Definition 2

(distance identifying function) A distance identifying functionf is an identifying function such that for every graph G and all vertices u, v and w of G with $u \ne v$:

($\alpha$)::

$f_G[w](u,v)$ is ${{\,\mathrm{false}\,}}$ when $d(w,u) = d(w,v)$.

We can now define the distance identifying sets and their associated meta-problem.

Definition 3

((f, r)-distance identifying set) Given a distance identifying function f and $r \in [\![\infty ]\!]$, a (f,r)-distance identifying set of a graph G is an r-dominating set of G that f-distinguishes G.

Distance Identifying Set

Input: A distance identifying function f and $r \in [\![\infty ]\!]$. A graph G.

Output: An (f, r)-distance identifying set of G of minimal size, if one exists.

Given a distance identifying function f and $r \in [\![\infty ]\!]$ as inputs of the meta-problem, the resulting problem is called (f, r)-Distance Identifying Set and denoted (f, r)-DIS. We also consider the parameterized version Distance Identifying Set(k).

An (f, r)-DIS problem is not necessarily NP-hard. For instance, let $\bot$ be the function such that $\bot _G[w](u,v)$ is false for every inputs. Then, $(\bot ,r)$-DIS can trivially be solved in constant time. We need to consider restrictions in which positive outputs appear.

Let $i \in \{0\} \cup [\![\infty ]\!]$, we suggest two criteria. First, we may restrain the range of a vertex to its i-neighborhood: a vertex should not distinguish two vertices if they do not lie in its i-neighborhood but it should always distinguish them whenever exactly one of them lies in that i-neighborhood. Reciprocally, we may ensure that a vertex could distinguish the vertices of its i-neighborhood: a vertex should distinguish a vertex belonging to its i-neighborhood from all the other vertices, assuming the distances are different. Formally, we have:

Definition 4

(i-local function) For $i \in \{0\} \cup [\![\infty ]\!]$, an i-local identifying function f is an identifying function such that for every graph G and all vertices u, v and w of G with $u \ne v$:

($\beta _1$)::

$f_G[w](u,v)$ is ${{\,\mathrm{true}\,}}$ when $d(w,u) \le i < d(w,v)$ or, symmetrically, $d(w,v) \le i < d(w,u)$.

($\beta _2$)::

$f_G[w](u,v)$ is ${{\,\mathrm{false}\,}}$ when $i < \min \{d(w,u), d(w,v)\}$.

Definition 5

(i-layered function) For $i \in \{0\} \cup [\![\infty ]\!]$, an i-layered identifying function f is an identifying function such that for every graph G and all vertices u, v, w of G with $u \ne v$:

($\gamma$)::

$f_G[w](u,v)$ is ${{\,\mathrm{true}\,}}$ when $\min \{d(w,u), d(w,v)\} \le i$ and $d(w,u) \ne d(w,v)$.

A problem (f, r)-DIS is said to be i-layered when the function f is i-layered, and it is said to be i-local when f is i-local and $r = i$. The problem is said local if it is i-local for an integer i. We observe that r-IC, r-LD and r-MD are r-local for each positive integer r, while MD is not local. However, for each $r \in [\![\infty ]\!]$, r-MD is r-layered, thus 1-layered. We also notice that r-LD is 0-layered while r-IC is not, which is their unique difference.

2.3 Detailed Computational Lower Bounds

Using the Distance Identifying Set meta-problem, we get the following lower bounds:

Theorem 4

For each 1-layered distance identifying functionfand every$r \in [\![\infty ]\!]{}$, the (f, r)-Distance Identifying Setproblem restricted to bipartite apex graphs isNP-hard, and does not admit an algorithm running in$2^{o(\sqrt{n})}$time under ETH.

Theorem 5

The local problems restricted to bipartite planar graphs areNP-hard, and do not admit an algorithm running in$2^{o(\sqrt{n})}$time under ETH.

Theorem 6

Letfbe a 1-layered distance identifying function,$r \in [\![\infty ]\!]{}$, andgbe aq-local distance identifying function. Both the (f, r)- and (g, q)-DISproblems areNP-hard, and do not admit:

• algorithms running in$2^{o(n)}$time, except if ETH fails,

• parameterized algorithms running in$2^{{\mathcal {O}}(k)} \cdot n^{{\mathcal {O}}(1)}$time, except if$W[2] = \mathsf{FPT}{}$.

even when restricted to bipartite graphs.

The parameter k denotes here the solution size of a relevant distance identifying set.

As a side result, Theorem 6 answers a question of Hartung and Nichterlein in [20]:

Corollary 1

Under ETH,Metric Dimensioncannot be solved in$2^{o(n)}$.

Finally, notice that the parameterized lower bound from Theorem 6 may be complemented by an elementary upper bound inspired from the kernel of IC and LD of size $2^k+k$:

Proposition 1

For everyr-local distance identifying functionf, the (f, r)-Distance Identifying Setproblem has a kernel of size$(r+1)^{k}+k$where k is the solution size. Therefore, iffis computable in polynomial time, enumerating the$\left( {\begin{array}{c}n\\ k\end{array}}\right)$sets ofkvertices of an input yields a naive algorithm running in time${\mathcal {O}}(n^{k+{\mathcal {O}}(1)}) \in {\mathcal{O}}^*(r^{(k^2)})$for (f, r)-DIS.

The proofs of the Theorems 4 to 6 will be given in Sect. 4.

3 The Supports of the Reductions for Distance Identifying Set

3.1 The Distance Identifying Graphs

Consider the associated graph $\phi (\varOmega ,{\mathcal {S}})$ as defined in Sect. 2.1. The differences between the Distance Identifying Set meta-problem and the dominating problem related to associated graphs actually raise two issues for a reduction based on these latter notions to be effective on Distance Identifying Set. First, contrarily to the dominating problem where a vertex may only discern its close neighborhood, the meta-problem may allow a vertex to discern further than its direct neighborhood. In that case, we cannot certify that a vertex $v^{\varOmega }_i$ does not distinguish a vertex $v^{{\mathcal {S}}}_j$ when $u_i$ is not in $S_j$, the adjacency not remaining a sufficient argument. Secondly, one may object that a vertex $v^{\varOmega }_i$ formally has to distinguish a vertex $v^{{\mathcal {S}}}_j$ from another vertex, but that distinguishing a single vertex is not defined.

To circumvent these problems, we suggest the following fix: rather than producing a single vertex for each $S_j \in {\mathcal {S}}$, the set $V_{\mathcal {S}}$ may contain two vertices $v^{{\mathcal {S}}}_j$ and ${\bar{v}}^{{\mathcal {S}}}_j$. Then, the role of $v^{\varOmega }_i$ would be to distinguish them if and only if $u_i \in S_j$. To ensure that the vertex $v^{\varOmega }_i$ distinguishes $v^{{\mathcal {S}}}_j$ and ${\bar{v}}^{{\mathcal {S}}}_j$ when $u_i \in S_j$, we may use the properties $(\beta _1)$ and $(\gamma )$ of Definitions 4 and 5 for the r-local and 1-layered problems, respectively. Precisely, when $u_i \in S_j$, $v^{\varOmega }_i$ should be at distance r to $v^{{\mathcal {S}}}_j$ (with $r = 1$ in the 1-layered cases) while ${\bar{v}}^{{\mathcal {S}}}_j$ should not be in the r-neighborhood of $v^{\varOmega }_i$. Similarly, to ensure that $v^{\varOmega }_i$ cannot distinguish $v^{{\mathcal {S}}}_j$ and ${\bar{v}}^{{\mathcal {S}}}_j$ when $u_i \not \in S_j$, we may use properties $(\alpha )$ or $(\beta _2)$ of Definitions 2 and 4. Hence, when $u_i \not \in S_j$, $v^{{\mathcal {S}}}_j$ should not be in the r-neighborhood of $v^{\varOmega }_i$, or $d(v^{\varOmega }_i,v^{{\mathcal {S}}}_j)$ and $d(v^{\varOmega }_i,{\bar{v}}^{{\mathcal {S}}}_j)$ should be equal.

That fix fairly indicates how to initiate the transformation of the associated graphs in order to deliver an equivalence between a hitting set formed by elements of $\varOmega$ and the vertices of a distance identifying set included in $V_\varOmega$. However, it is clearly not sufficient since we also have to distinguish the couples of vertices of $V_\varOmega$ for which nothing is required. To solve that problem, we suggest to append to each vertex of the associated graph a copy of some gadget with the intuitive requirement that the gadget is able to distinguish the close neighborhood of its vertices from the whole graph. We introduce the notion of B-extension:

Definition 6

(B-extension) Let $H = (V_H,E_H)$ be a connected graph, and $B \subseteq V_H$. An induced supergraph $G = (V_G,E_G)$ is said to be a B-extension of H if it is connected and for every vertex v of $V_{G \setminus H}$, the set $N(v) \cap V_H$ is either equal to $\emptyset$ or B.

A vertex v of $V_{G \setminus H}$ such that $N(v) \cap V_H = B$ is said to be B-adjacent. The B-extensions of H such that $V_{G \setminus H}$ contains exactly a B-adjacent vertex or two B-adjacent vertices but not connected to each other are called theB-single-extension and theB-twin-extension of H, respectively.

Here, the “border” B makes explicit the connections between a copy of a gadget H and a vertex outside the copy. In particular, a B-single-extension is formed by a gadget with its related vertex $v^{\varOmega }_i$, while a B-twin-extension contains a gadget with its two related vertices $v^{{\mathcal {S}}}_j$ and ${\bar{v}}^{{\mathcal {S}}}_j$. Piecing all together, we may adapt the associated graphs to the meta-problem:

Definition 7

((H, B, r)-distance identifying graph) Let $(\varOmega = \{u_i \; | \; i \in [\![n]\!]\},{\mathcal {S}} = \{S_i \; | \; i \in [\![m]\!]\})$ be an instance of Hitting Set. Let H be a connected graph, B a subset of its vertices, and r a positive integer. The (H, B, r)-distance identifying graph$\varPhi [H,B,r](\varOmega ,{\mathcal {S}})$ is as follows.

• for each $i \in [\![n]\!]$, the graph $\varPhi [H,B,r](\varOmega ,{\mathcal {S}})$ contains as induced subgraph a copy $H^{\varOmega }_i$ of H together with a $B^{\varOmega }_i$-adjacent vertex $v^{\varOmega }_i$, where $B^{\varOmega }_i$ denotes the copy of B.

• similarly, for each $j \in [\![m]\!]$, the graph $\varPhi [H,B,r](\varOmega ,{\mathcal {S}})$ contains a copy $H^{{\mathcal {S}}}_j$ of H together with two $B^{{\mathcal {S}}}_j$-adjacent vertices $v^{{\mathcal {S}}}_j$ and ${\bar{v}}^{{\mathcal {S}}}_j$ (the latter vertices are not adjacent) and where $B^{{\mathcal {S}}}_j$ denotes the copy of B.

• finally, for each $S_j \in {\mathcal {S}}$ and each $u_i \in S_j$, $v^{\varOmega }_i$ is connected to $v^{{\mathcal {S}}}_j$ by a path of $r-1$ vertices denoted $l^k_{i,j}$ with $d(v^{\varOmega }_i,l^k_{i,j}) = k$ for each $k \in [\![r-1]\!]$.

When the problem is not local, we prefer the following identifying graph:

Definition 8

((H, B)-apex distance identifying graph) An (H, B)-apex distance identifying graph$\varPhi ^*[H,B](\varOmega ,{\mathcal {S}})$ is the union of an (H, B, 1)-distance identifying graph with an additional vertex a called apex such that:

• for each $u_i \in \varOmega$, the apex a is $B^\varOmega _i$-adjacent to $H^\varOmega _i$, where $B^{\varOmega }_i$ (resp. $H^\varOmega _i$) denotes the copy of B (resp. H).

• for each $S_j \in {\mathcal {S}}$, the apex a is adjacent to $v^{{\mathcal {S}}}_j$ and ${\bar{v}}^{{\mathcal {S}}}_j$.

See Fig. 2 for an example of an (H, B, r)-distance identifying graph (on the left) and an example of (H, B)-apex distance identifying graph (on the right).

Proposition 2

Given an instance$(\varOmega ,{\mathcal {S}})$ofPlanarHitting Setwhere$|\varOmega | = n$, $|{\mathcal {S}}| = m$, the graphs$G = \varPhi [H,B,r](\varOmega ,{\mathcal {S}})$and$G' = \varPhi ^*[H,B](\varOmega ,{\mathcal {S}})$

• are connected and have size bounded by$(|H|+2r)(n+m)$, (with$r = 1$ for $G'$),

• may be built in polynomial time in their size,

• are bipartite if theB-single extension ofHis bipartite,

• are respectively planar and an apex graph if theB-twin-extension ofHis planar.

Proof

The graph G is formed by the union of nB-single-extensions of H, mB-twin-extensions of H and all the possible paths of $r-1$ vertices. As $\phi (\varOmega ,{\mathcal {S}})$ is a bipartite planar graph, the Euler formula implies that the number of paths is bounded by $2(n+m)-4$. We conclude that the number of vertices of G is bounded by:

$$\begin{aligned}n(|H|+1)+m(|H|+2)+(r-1)(2(n+m)-4) = (|H|+2r)(n+m)-n-4(r-1)\end{aligned}$$

Furthermore, it is clear that G is connected if and only if the associated graph $\phi (\varOmega ,{\mathcal {S}})$ is connected. Additionally, we may consider that $\phi (\varOmega ,{\mathcal {S}})$ is connected since it is a property decidable in polynomial time, and that the instances corresponding to the distinct connected components of $\phi (\varOmega ,{\mathcal {S}})$ may be considered independently.

Finally, all the other items of the proposition are direct by construction. $\square$

Fig. 2

An (H, B, 3)-distance identifying graph and an (H, B)-apex distance identifying graph built on the planar instance formed by $\varOmega = \{1,2,3,4\}$ and ${\mathcal {S}} = \{\{1,2\},\{2,3,4\}\}$

Having defined the (apex) distance identifying graphs, the main effort to obtain generic reduction from PlanarHitting Set is done. We now define relevant gadgets:

Definition 9

((f, r)-gadgets) Let f be a distance identifying function and $r \in [\![\infty ]\!]$. Let $H = (V_H,E_H)$ be a connected graph, and B, C be two subsets of $V_H$. We say that the triple (H, B, C) is a (f, r)-gadget if for every B-extension G of H:

$(p_h)$:

Cf-distinguishes $V_H$ and $V_G$.

$(p_b)$:

Cf-distinguishes $N_B$ and $V_{G \setminus H} \setminus N_B$, where $N_B$ is the set of B-adjacent vertices of G.

$(p_d)$:

C is an r-dominating set of $G[V_H \cup N_B]$.

$(p_s)$:

For all (f, r)-distance identifying sets S of G, $|C| \le |S \cap V_H|$.

Definition 10

(local gadgets) An (f, r)-gadget is a local gadget, if f is an r-local identifying function with $r \ne \infty$, and

$(p_l)$:

for every $k \in [\![r]\!]$, there exists $c \in C$ such that $d(c,B) = k-1$.

Consistently, we say that an (f, r)-gadget (H, B, C) is bipartite if the B-single-extension of H is bipartite, and that it is planar if the B-twin-extension of H is planar.

Theorem 7

Let$(\varOmega ,{\mathcal {S}})$be an instance ofHitting Setsuch that$|\varOmega | = n > 1$, $|{\mathcal {S}}| = m$. Let (H, B, C) be an (f, r)-gadget for a 1-layered identifying functionfand let$(H',B',C')$be a local (g, q)-gadget. The following propositions are equivalent:

• There exists a hitting set of${\mathcal {S}}$of sizek.

• There exists an (f, r)-distance identifying set of$\varPhi ^*[H,B](\varOmega ,{\mathcal {S}})$of size$k+|C|(n+m)$.

• There exists a (g, q)-distance identifying set of$\varPhi [H',B',q](\varOmega ,{\mathcal {S}})$of size$k+|C'|(n+m)$.

Proof

We start by focusing on the equivalence between the first and second items.

Suppose first that P is a hitting set of $(\varOmega ,{\mathcal {S}})$ of size k. By denoting $C^\varOmega _i$ and $C^{\mathcal {S}}_j$ the copies of C associated to the copies $H^\varOmega _i$ and $H^{\mathcal {S}}_j$ of H, we suggest the following set I of size $k+|C|(n+m)$ as an (f, r)-distance identifying set of $G = \varPhi ^*[H,B](\varOmega ,{\mathcal {S}})$:

$$\begin{aligned} I = \{ v^\varOmega _i :u_i \in P \} \; \cup \underset{i \, \in \, [\![n]\!]}{\bigcup } C^\varOmega _i \cup \underset{j \, \in \, [\![m]\!]}{\bigcup } C^{\mathcal {S}}_j. \end{aligned}$$

Recall that by construction, G is a $B^\varOmega _i$-extension of $H^\varOmega _i$ (respectively $B^{\mathcal {S}}_j$-extension of $H^{\mathcal {S}}_j$) for any $i \in [\![n]\!]$ (respectively $j \in [\![m]\!]$). This directly implies that I is an r-dominating set of G. Indeed, the condition $(p_d)$ of Definition 9 implies that $C^\varOmega _i$ (respectively $C^{\mathcal {S}}_j$) r-dominates $H^\varOmega _i$ plus $v^\varOmega _i$ (respectively of $H^{\mathcal {S}}_j$ plus $v^{\mathcal {S}}_j$, ${\bar{v}}^{\mathcal {S}}_j$). The remaining apex is also r-dominated by any $C^\varOmega _i$, as it is $B^\varOmega _i$-adjacent for every $i \in [\![n]\!]$.

We now have to show that If-distinguishes G. We begin with the vertices of the gadget copies because the condition $(p_h)$ implies that $C_i^\varOmega \subseteq I$f-distinguishes the vertices of $H_i^\varOmega$ and G for every $i \in [\![n]\!]$, and If-distinguishes the vertices of $H_j^{\mathcal {S}}$ and G for every $j \in [\![m]\!]$. Thereby, we only have to study the vertices of the form $v^\varOmega _i$, $v^{\mathcal {S}}_j$, ${\bar{v}}^{\mathcal {S}}_j$, and the apex a (there is no vertex of the form $l^k_{i,j}$ in an apex distance identifying graph). To distinguish them, we use the condition $(p_b)$. Recall that $n > 1$. Then, for each distinct $i,i' \in [\![n]\!]$, we have:

• $v^\varOmega _i$ is $B^\varOmega _i$-adjacent but not $B^\varOmega _{i'}$-adjacent,

• a is both $B^\varOmega _i$-adjacent and $B^\varOmega _{i'}$-adjacent,

• a vertex of the form $v^{\mathcal {S}}_j$ or ${\bar{v}}^{\mathcal {S}}_j$ is neither $B^\varOmega _i$-adjacent nor $B^\varOmega _{i'}$-adjacent.

Enumerating the relevant i and $i'$, we deduce that every couple of vertices is distinguished except when they are both of the form $v^{\mathcal {S}}_j$ or ${\bar{v}}^{\mathcal {S}}_{j'}$ for $j,j' \in [\![m]\!]$. But we may distinguish $v^{\mathcal {S}}_j$ or ${\bar{v}}^{\mathcal {S}}_{j'}$ for distinct $j,j'$ by applying $(p_b)$ on $H^{\mathcal {S}}_j$.

It remains to distinguish $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_{j}$ for $j \in [\![m]\!]$. We now use the fact that P is a hitting set for $(\varOmega ,{\mathcal {S}})$. By definition of a hitting set, for any set $S_j \in {\mathcal {S}}$, there exists a vertex $u_i \in P$ such that $u_i \in S_j$. We observe that $d(v^\varOmega _i,v^{\mathcal {S}}_j) = 1 < d(v^\varOmega _i,{\bar{v}}^{\mathcal {S}}_j)$ by construction of G and that $v^\varOmega _i \in I$ by definition of I. Since f is 1-layered, If-distinguishes $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$.

In the other direction, assume that I is a distance identifying set of G of size $k+|C|(n+m)$. As every set of ${\mathcal {S}}$ is not empty, we may define a function $\varphi : [\![m]\!] \rightarrow [\![n]\!]$ such that $u_{\varphi (j)} \in S_j$.

We suggest the following set P as a hitting set of ${\mathcal {S}}$ of size at most k:

$$\begin{aligned} P = \{ u_i \in \varOmega \; | \; v^\varOmega _i \in I \} \cup \{ u_{\varphi (j)} \in \varOmega \; | \; v^{\mathcal {S}}_j \in I \text { or } {\bar{v}}^{\mathcal {S}}_j \in I\} \end{aligned}$$

We claim that the only vertices that may f-distinguish $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ are themselves and the vertices $v^\varOmega _i$ such that $u_i \in S_j$. To prove so, we apply propriety $(\alpha )$ of Definition 2:

• The apex a verifies $d(a,v^{\mathcal {S}}_j) = 1 = d(a,{\bar{v}}^{\mathcal {S}}_j)$.

• A vertex $v^\varOmega _i$ such that $u_i \not \in S_j$ verifies $d(v^\varOmega _i,v^{\mathcal {S}}_j) = 3 = d(v^\varOmega _i,{\bar{v}}^{\mathcal {S}}_j)$.

• A vertex v of $H^\varOmega _i$ verifies $d(v,v^{\mathcal {S}}_j) = 2+d(v,B^\varOmega _i) = d(v,{\bar{v}}^{\mathcal {S}}_j)$.

• A vertex v of $H^{\mathcal {S}}_{j'}$ with $j \ne j'$ verifies $d(v,v^{\mathcal {S}}_j) = 3+ d(v,B^{\mathcal {S}}_j) = d(v,{\bar{v}}^{\mathcal {S}}_j)$.

• Both $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ are $B^{\mathcal {S}}_j$-adjacent, so they are at the same distance of any vertex of $H^{\mathcal {S}}_j$.

We deduce that $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ are f-distinguished only if either one on them belongs to I (in that case $u_{\varphi (j)} \in P \cap S_j$) or there exists $v^\varOmega _i \in I$ such that $u_i \in S_j$ (and then $u_i \in P \cap S_j$).

It remains to show that $|P| \le k$. By the condition $(p_s)$ of Definition 9, we know that $|I \cap V_{H^\varOmega _i}| \ge |C^\varOmega _i|$ and $|I \cap V_{H^{\mathcal {S}}_j}| \ge |C^{\mathcal {S}}_j|$ for any $i \in [\![n]\!]$ and $j \in [\![m]\!]$, implying

$$\begin{aligned} \begin{array}{rllll} k &{} =|I| - |C|(n+m) \\ &{} \ge \sum \limits _{i \in [n]} |I \cap \{v^\varOmega _i\}| &{} +&{} \sum \limits _{j \in [m]} |I \cap \{v^{\mathcal {S}}_j ,{\bar{v}}^{\mathcal {S}}_j\}| \\ &{} \ge \sum \limits _{v^\varOmega _i \in I} 1 &{} +&{} \sum \limits _{ I \cap \{v^{\mathcal {S}}_j,{\bar{v}}^{\mathcal {S}}_j\} \ne \emptyset } 1 &{}= |P|\\ \end{array} \end{aligned}$$

Now, we prove the equivalence between the first and third items. Consider a q-local distance identifying function g, a local (g, q)-gadget (H, B, C) and an instance $(\varOmega ,{\mathcal {S}})$ of PlanarHitting Set such that $|\varOmega | = n$, $|{\mathcal {S}}| = m$. We denote the copies of H as $H_i^\varOmega$ or $H_j^{\mathcal {S}}$, the copies of C as $C_i^\varOmega$ and $C_j^{\mathcal {S}}$, and the copies of B as $B_i^\varOmega$ and $B_j^{\mathcal {S}}$ for any $i \in [\![n]\!]$ and $j \in [\![m]\!]$.

In the first direction, suppose that P is a hitting set of $(\varOmega ,{\mathcal {S}})$ of size k, the (g, q)-distance identifying set I of $G = \varPhi [H,B,q](\varOmega ,{\mathcal {S}})$ is defined identically as in the equivalence of the first and second items of the current theorem:

$$\begin{aligned} I = \{ v_i^\varOmega :u_i \in P\} \cup \bigcup _{i \in [n]} C_i^\varOmega \cup \bigcup _{j \in [m]} C_j^{\mathcal {S}}\end{aligned}$$

Using conditions $(p_d)$ and $(p_l)$ of Definitions 9 and 10, I is a q-dominating set of G. Indeed, by $(p_d)$ every vertex belonging to a copy of the gadget is q-dominated. Additionally, every vertex outside of the copies of the gadgets is at distance at most q of a copy by construction, but there exists a vertex $b \in B \cap C$ (so a relevant copy in I) by $(p_l)$.

To prove that Ig-distinguishes G, the strategy is differing from the previous equivalence only on the $l_{i,j}^k$ vertices and when distinguishing $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ as we will see.

Recall that by construction, G is a $B^\varOmega _i$-extension of $H^\varOmega _i$ (respectively $B^{\mathcal {S}}_j$-extension of $H^{\mathcal {S}}_j$) for any $i \in [\![n]\!]$ (respectively $j \in [\![m]\!]$). Distinguishing the vertices of the gadget copies is easy, as the condition $(p_h)$ implies that $C_i^\varOmega \subseteq I$g-distinguishes the vertices of $H_i^\varOmega$ and G for every $i \in [\![n]\!]$, and similarly Ig-distinguishes the vertices of $H_j^{\mathcal {S}}$ and G for every $j \in [\![m]\!]$.

Thereby, we only have to study the vertices of the form $v^\varOmega _i$, $v^{\mathcal {S}}_j$, ${\bar{v}}^{\mathcal {S}}_j$, and the vertices $l_{i,j}^k$.

To distinguish them, we mainly use the condition $(p_b)$. We observe that for each distinct $i,i' \in [\![n]\!]$ (they exist as $n > 1$) :

• The vertex $v^\varOmega _i$ is $B^\varOmega _i$-adjacent but not $B^\varOmega _{i'}$-adjacent.

• For every $j \in [m]$, $v^{\mathcal {S}}_j$ or ${\bar{v}}^{\mathcal {S}}_j$ is neither $B^\varOmega _i$-adjacent nor $B^\varOmega _{i'}$-adjacent.

• For every $i \in [n]$, $j \in [m]$ and $k \in [r-1]$, $l_{i,j}^k$ is neither $B^\varOmega _i$-adjacent nor $B^\varOmega _{i'}$-adjacent.

Thus Ig-distinguishes $v_i^\varOmega$ and G.

As the vertices of form $l_{i,j}^k$ are the only vertices to belong to both the q-neighbourhood of $B_i^\varOmega$ and $B_j^{\mathcal {S}}$, and as the vertices $l_{i,j}^k$ and $l_{i,j}^{k'}$ with $k < k'$ are g-distinguished by the guaranteed vertex $c \in C_i^\varOmega$ such that $d(c,B_i^\varOmega ) = q-k-1$, Ig-distinguishes $l_{i,j}^k$ and G for every relevant i, j and k.

It remains to distinguish $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_{j'}$ for $j,j' \in [\![m]\!]$. If j and $j'$ are distinct we may use $(p_b)$ on the copy $H_j^{\mathcal {S}}$ of the gadget H. We may assume $j = j'$. We now use the fact that P is a hitting set for $(\varOmega ,{\mathcal {S}})$. By definition of a hitting set, for any set $S_j \in {\mathcal {S}}$, there exists a vertex $u_i \in P$ such that $u_i \in S_j$. We observe that $d(v^\varOmega _i,v^{\mathcal {S}}_j) = q < d(v^\varOmega _i,{\bar{v}}^{\mathcal {S}}_j)$ (when $u_i \in S_j$) by construction of G and I indeed g-distinguishes $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ because g is q-local.

In the other direction, assume that I is a distance identifying set of G of size $k+|C|(n+m)$. The hitting set P may now depend on $l_{i,j}^k$. Let define $L_i = \{v^\varOmega _i\} \cup \{l_{i,j}^k \; | \; k \in [\![r-1]\!] \text { and } u_i \in S_j\}$. As every set of S is not empty, we may define a function $\varphi : [\![m]\!] \rightarrow [\![n]\!]$ such that $u_{\varphi (j)} \in S_j$. We suggest the following set P as a hitting set of ${\mathcal {S}}$ of size at most k:

$$\begin{aligned} P = \{ u_i \in \varOmega \; | \; I \cap L_i \ne \emptyset \} \cup \{ u_{\varphi (j)} \in \varOmega \; | \; v^{\mathcal {S}}_j \in I \text { or } {\bar{v}}^{\mathcal {S}}_j \in I\} \end{aligned}$$

Consider $j \in [\![m]\!]$, let us show that the only vertices that may g-distinguish the couple $(v^{\mathcal {S}}_j, {\bar{v}}^{\mathcal {S}}_j)$ are themselves and the vertices from $L_i$ (and not only $v_i^\varOmega$) such that $u_i \in S_j$. Every vertex from $H_j^{\mathcal {S}}$ is at the same distance to $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ and thus cannot g-distinguishes them because of the distance property $(\alpha )$. Every vertex not in $H_j^{\mathcal {S}}$, not in $L_i$ for every $i \in [\![n]\!]$ such that $u_i \in S_j$ and different from $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ is at distance at least $q+1$ of the two latter vertices. Thus, because of the propriety $(\beta _2)$ of Definition 4 (a vertex cannot distinguish two vertices outside of its q-neighbourhood) any of these vertices does not q-distinguish $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$. We deduce that $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ are g-distinguished if and only if either one of them belongs to I (in that case $u_{\varphi (j)} \in P \cap S_j$) or there exists $i \in [\![n]\!]$ such that $u_i \in S_j$ and $I \cap L_i \ne \emptyset$.

The proof that $|P| \le k$ is provided by $(p_s)$, we know that $|I \cap V_{H^\varOmega _i}| \ge |C^\varOmega _i|$ and $|I \cap V_{H^{\mathcal {S}}_j}| \ge |C^{\mathcal {S}}_j|$ for any $i \in [\![n]\!]$ and $j \in [\![m]\!]$. Considering the following partition of I

$$\begin{aligned} { I} = \Big (\bigsqcup \limits _{i \in [\![n]\!]} (I \cap H_i^\varOmega ) \Big ) \bigsqcup \Big (\bigsqcup \limits _{j \in [\![m]\!]} (I \cap H_j^{\mathcal {S}}) \Big ) \bigsqcup \Big ( \bigsqcup \limits _{i \in [\![n]\!]} (I \cap L_i) \Big ) \bigsqcup \Big ( \bigsqcup \limits _{j \in [\![m]\!]} (I \cap \{ v_j^{\mathcal {S}}, {\bar{v}}_j^{\mathcal {S}}) \}\Big ) \end{aligned}$$

We get

$$\begin{aligned} \begin{array}{rlllllll} |I| &{} \ge |C|(n+m) &{} + &{} {\displaystyle \sum _{i \in [\![n]\!]} |I \cap L_i|}&{} + &{} {\displaystyle \sum _{j \in [\![m]\!]} |I \cap \{v^{\mathcal {S}}_j ,{\bar{v}}^{\mathcal {S}}_j\}|} \\ &{} \ge |C|(n+m) &{} +&{} {\displaystyle \sum _{I \cap L_i \not = \emptyset } 1} &{} +&{} {\displaystyle \sum _{ I \cap \{v^{\mathcal {S}}_j,{\bar{v}}^{\mathcal {S}}_j\} \ne \emptyset } 1} \\ &{} = |C|(n+m) + |P| \end{array} \end{aligned}$$

Because $|I| = k + |C|(n+m)$, we conclude that $|P| \le k$.

Obviously, the second and third items are equivalent since they are both equivalent to the first item, which concludes the proof. $\square$

3.2 Binary Compression of Gadgets

The Theorem 7 is a powerful tool to get reductions, in particular in the planar cases. However, the number of involved gadgets does not allow to use Theorem 1. This limitation is due to the uses of a gadget per vertex to identify in the distance identifying graphs. Using power set, we may obtain a better order: given k gadgets, we may identify $2^k-1$ vertices (we avoid to identify a vertex with the empty subset of gadgets). Thus, we will consider binary representations of integers as sequences of bits, with weakest bit at last position. For a positive integer n, we define the integer $b_{n} = 1+\lfloor \log _2(n) \rfloor$ and introduce a new graph:

Definition 11

((H, B, r)-compressed graph) Let $(\varOmega , {\mathcal {S}})$ be an instance of Hitting Set, with $\varOmega = \{u_i \; | \; i \in [\![n]\!]\}$ and ${\mathcal {S}} = \{S_i \; | \; i \in [\![m]\!]\}$. Let H be a connected graph, B be a subset of its vertices, and r be a positive integer. The (H,B,r)-compressed graph$\varPsi [H,B,r](\varOmega ,{\mathcal {S}})$ is defined as follows. $\varPsi [H,B,r](\varOmega ,{\mathcal {S}})$ contains as induced subgraphs $b_{n+1}$ copies of H denoted $H^{\varOmega }_i$ for $i \in [\![b_{n+1}]\!]$ and $b_{m}$ other copies of H denoted $H^{{\mathcal {S}}}_j$ for $j \in [\![b_{m}]\!]$. Then:

• For each $j \in [\![m]\!]$, we add two non-adjacent vertices $v^{{\mathcal {S}}}_j$ and ${\bar{v}}^{{\mathcal {S}}}_j$. They are $B^{{\mathcal {S}}}_k$-adjacent for each $k \in [\![b_{m}]\!]$ such that the $k^{th}$ bit of the binary representation of j is 1.

• For each $i \in [\![n]\!]$, we add r vertices denoted $l_i^{j-1}$ with $j \in [\![r]\!]$ to form a fresh path such that $d(v^{\varOmega }_i,l_i^{j-1}) = j-1$ where $v^{\varOmega }_i = l_i^0$. We make $v^{\varOmega }_i$$B^{\varOmega }_k$-adjacent for each $k \in [\![b_{n+1}]\!]$ such that the $k^{th}$ bit of the binary representation of i is 1.

• For each $S_j \in {\mathcal {S}}$ and each $u_i \in S_j$, we add the edge $(l_i^{r-1},v^{{\mathcal {S}}}_j)$.

• We add r vertices denoted $a^{j-1}$ with $j \in [\![r]\!]$ to form the induced path such that $d(a^0,a^{j-1}) = j-1$. The vertex $a^0$ is $B^{\varOmega }_k$-adjacent for every $k \in [\![b_{n+1}]\!]$, and we add the edges $(a^{r-1},v^{{\mathcal {S}}}_j)$ and $(a^{r-1},{\bar{v}}^{{\mathcal {S}}}_j)$ for each $j \in [\![m]\!]$.

By definition of $b_{n+1}$, for every $i \in [\![n]\!]$, one of the last $b_{n+1}$ bits of the binary representation of i is 0. So, $a^0$ has a distinct characterization in the power set formed by the gadgets $H^\varOmega _i$. See Fig. 3 for an example of (H, B, r)-compressed graph.

Fig. 3

The (H, B, 2)-compressed graph where $\varOmega = \{1,2,3,4\}$ and ${\mathcal {S}} = \{\{1,2\}, \{2,3,4\}\}$

Proposition 3

The graph$G = \varPsi [H,B,r](\varOmega ,{\mathcal {S}})$built on an instance$(\varOmega ,{\mathcal {S}})$ofHitting Set:

• is connected and has size at most$|H|(b_{n+1}+b_{m})+r(n+1)+2m$, where$|\varOmega | = n$and$|{\mathcal {S}}| = m$,

• may be built in polynomial time in its size, and

• is bipartite if theB-single extension ofHis bipartite.

Proof

The graph G is formed by the union of $b_{n+1} + b_{m}$ copies of H, one vertex per variable, two vertices per clause, n paths of $r-1$ vertices and one path of size r. Thus, in total, the number of vertices is

$$\begin{aligned}(b_{n+1}+b_{m})|H|+n+2m+n(r-1)+r= |H|(b_{n+1}+b_{m})+r(n+1)+2m\end{aligned}$$

Finally, the two last items of the proposition are also direct by construction. $\square$

Theorem 8

Let$(\varOmega ,{\mathcal {S}})$be an instance ofHitting Setsuch that$|\varOmega | = n$, $|{\mathcal {S}}| = m$. Let (H, B, C) be an (f, r)-gadget for a 1-layered identifying functionfand let$(H',B',C')$be a local (g, q)-gadget. The following propositions are equivalent:

• There exists a hitting set of${\mathcal {S}}$of sizek.

• There exists an (f, r)-distance identifying set of the graph$\varPsi [H,B,1](\varOmega ,{\mathcal {S}})$which size is$k+|C|(b_{n+1}+b_{m})$.

• There exists a (g, q)-distance identifying set of the graph$\varPsi [H',B',q](\varOmega ,{\mathcal {S}})$which size is$k+|C'|(b_{n+1}+b_{m})$.

Proof

Suppose again that P is a hitting set of $(\varOmega ,{\mathcal {S}})$ of size k. By denoting $C^\varOmega _i$ and $C^{\mathcal {S}}_j$ the copy of C (respectively $C'$) associated to the copy $H^\varOmega _i$ and $H^{\mathcal {S}}_j$ of H (respectively $H'$), we suggest the following set I of size $k+|C|(b_{n+1}+b_{m})$ as an (f, r)-distance identifying set of $G = \varPsi [H,B,1](\varOmega ,{\mathcal {S}})$ (respectively (g, q)-distance identifying set of $G' = \varPsi [H',B',q](\varOmega ,{\mathcal {S}}))$:

$$\begin{aligned} I = \{ v^\varOmega _i :u_i \in P \} \; \cup \bigcup _{i \, \in \, [\![b_{n+1}]\!]} C^\varOmega _i \; \cup \bigcup _{j \, \in \, [\![b_{m+1}]\!]} C^{\mathcal {S}}_j. \end{aligned}$$

By construction, G is a $B^\varOmega _i$-extension of $H^\varOmega _i$ (respectively $B^{\mathcal {S}}_j$-extension of $H^{\mathcal {S}}_j$) for any $i \in [\![b_{n+1}]\!]$ (respectively $j \in [\![b_{m}]\!]$). This directly implies that I is an r-dominating set of G (respectively q-dominating set $G'$).

We only have to show that If-distinguishes G (respectively g-distinguishes $G'$). Distinguishing the vertices of the gadget copies is still easy using the first item of Definition 9. Thereby, we only have to study the vertices of the form $v^{\mathcal {S}}_j$, ${\bar{v}}^{\mathcal {S}}_j$, $l^k_i$, and $a^k$. To distinguish them, we mainly use the second item of Definition 9 together with the characteristic function of the power set of the gadgets. We deduce that every couple of vertices is distinguished except when the two vertices are of the form $v^{\mathcal {S}}_j$ or ${\bar{v}}^{\mathcal {S}}_{j}$ for $j \in [\![m]\!]$ (or if they are both of the form $a^k$ or $l^k_i$ for $k \in [\![q-1]\!]$ for $G'$, G not containing such vertices).

Let us now distinguish $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_{j}$ for $j \in [\![m]\!]$. To do so, we use the fact that P is a hitting set for $(\varOmega ,{\mathcal {S}})$. By definition of a hitting set, for any set $S_j \in {\mathcal {S}}$, there exists a vertex $u_i \in P$ such that $u_i \in S_j$. We observe that $d(v^\varOmega _i,v^{\mathcal {S}}_j) = 1 < d(v^\varOmega _i,{\bar{v}}^{\mathcal {S}}_j)$ by construction of G (respectively $d(v^\varOmega _i,v^{\mathcal {S}}_j) = q < d(v^\varOmega _i,{\bar{v}}^{\mathcal {S}}_j)$ by construction of $G'$) and that $v^\varOmega _i \in I$ by definition of I. Since f is 1-layered (respectively g is q-local), If-distinguishes and g-distinguishes $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$.

For $G'$, it remains to distinguish $a^k$ and $l^k_i$ for $k \in [\![q-1]\!]$ and $i \in [\![n]\!]$. We recall that in a (g, q)-local gadget $(H',B',C')$, there exists $c \in C'$ such that $d(c,B') = k-1$ for each $k \in [\![q]\!]$. Then we may use the characteristic function of the power set together with property $(\beta _1)$ of a q-local function to distinguish them.

In the other direction, assume that I is an (f, r)-distance identifying set of G of size $k+|C|(b_{n+1}+b_{m})$ (respectively (g, q)-distance identifying set of G of size $k+|C'|(b_{n+1}+b_{m})$). As every set of ${\mathcal {S}}$ is not empty, we may define a function $\varphi : [\![m]\!] \rightarrow [\![n]\!]$ such that $u_{\varphi (j)} \in S_j$.

We suggest the following set P as a hitting set of ${\mathcal {S}}$ of size at most k:

$$\begin{aligned} P = \{ u_i \in \varOmega \; | \; l^k_i \in I \text { for any } k \in [\![r-1]\!] \} \cup \{ u_{\varphi (j)} \in \varOmega \; | \; v^{\mathcal {S}}_j \in I \text { or } {\bar{v}}^{\mathcal {S}}_j \in I\} \end{aligned}$$

The size of P is ensured by the fourth item of Definition 9 of a gadget.

We claim that the only vertices that may f-distinguish the couple $(v^{\mathcal {S}}_j, {\bar{v}}^{\mathcal {S}}_j)$ are themselves and the vertices of form $l^k_i$ such that $u_i \in S_j$. To prove so, we apply property $(\alpha )$ from Definition 2 (a vertex cannot distinguish two vertices at the same distance from it) on the following enumeration on G:

• The vertices $a^k$ verify $d(a^k,v^{\mathcal {S}}_j) = r-k = d(a^k,{\bar{v}}^{\mathcal {S}}_j)$ for each $k \in [\![r-1]\!]$.

• A vertex $l^k_i$ such that $u_i \not \in S_j$ verifies $d(l^k_i,v^{\mathcal {S}}_j) = 2+r-k = d(l^k_i,{\bar{v}}^{\mathcal {S}}_j)$ because of $a^{r-1}$.

• A vertex v of $H^\varOmega _i$ verifies $d(v,v^{\mathcal {S}}_j) = d(v,B^\varOmega _i)+1+r = d(v,{\bar{v}}^{\mathcal {S}}_j)$ because of the path formed by the vertices of form $a^k$.

• A vertex v of $H^{\mathcal {S}}_{j'}$ with $j \ne j'$ verifies $d(v,v^{\mathcal {S}}_j) = d(v,B^{\mathcal {S}}_{j'})+3 = d(v,{\bar{v}}^{\mathcal {S}}_j)$ because of $a^{r-1}$.

• Both $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ are $B^{\mathcal {S}}_j$-adjacent, so they are at the same distance of any vertex of $H^{\mathcal {S}}_j$.

The enumeration on $G'$ is identical when replacing r by q. We deduce that $v^{\mathcal {S}}_j$ and ${\bar{v}}^{\mathcal {S}}_j$ are f-distinguished if and only if either one on them belongs to I (in that case $u_{\varphi (j)} \in P \cap S_j$) or if there exists $l^k_i \in I$ such that $u_i \in S_j$ and $k+1 \in [\![r]\!]$ (and then $u_i \in P \cap S_j$). $\square$

4 On Providing Gadgets to Establish Generic Reductions

In this section, we finalize the reductions by furnishing some gadgets and combining them with the suitable theorems and propositions from Sect. 3. The existence of the gadgets rely on the following tool lemma:

Lemma 1

(Twins Lemma) Letxandybe two vertices of a graphGsuch that$N(x) = N(y)$. Then any distance identifying set ofGcontains eitherxory.

Proof

Because $N(x) = N(y)$, for every vertex u of G, if $u \not \in \{x,y\}$, then $d(u,x) = d(u,y)$. Thus, by property $(\alpha )$ of a distance identifying set, u may distinguish x and y if and only if $u \in \{x,y\}$, implying that a distance identifying set must contain either x or y. $\square$

The gadgets are defined as follows:

Definition 12

(The 1-layered gadget) Let H be the bipartite planar graph such that:

• Its ten vertices are denoted b, ${\bar{b}}$, $u_1$, ${\bar{u}}_1$, $u_2$, ${\bar{u}}_2$, $v_1$, ${\bar{v}}_1$, $v_2$ and ${\bar{v}}_2$.

• The vertices $u_1, u_2, {\bar{u}}_1$ and ${\bar{u}}_2$ form a cycle as well as the vertices $v_1, v_2, {\bar{v}}_1$ and ${\bar{v}}_2$.

• The vertices b and ${\bar{b}}$ are adjacent to $u_1$, ${\bar{u}}_1$, $v_1$ and ${\bar{v}}_1$.

We define the sets $B = \{b,{\bar{b}}\}$ and $C = \{b, u_1, u_2, v_1, v_2\}$.

The triple (H, B, C) is called the 1-layered gadget (see Fig. 4).

Fig. 4

The 1-layered gadget (H, B, C). The set C contains the colored vertices and the set B contains the square vertices

Proposition 4

The 1-layered gadget is a bipartite planar (f, r)-gadget for any 1-layered distance identifying functionfand$r \in [\![\infty ]\!]$.

Proof

We have to check the four conditions to be an (f, r)-gadget. Consider a B-extension G of H. Clearly, $(p_d)$ is satisfied as C is even a 1-dominating set of $V_H \cup N_B$. The condition $(p_s)$ is also easily verified using the Twins Lemma 1 on the distinct pairs $(b,{\bar{b}})$, $(u_1,{\bar{u}}_1)$, $(u_2,{\bar{u}}_2)$, $(v_1,{\bar{v}}_1)$ and $(v_2,{\bar{v}}_2)$. To prove $(p_h)$ and $(p_b)$, we only have to study vertices not belonging to C. Remark that:

• The vertex ${\bar{u}}_1$ is the only one outside of C that is adjacent to $u_2$.

• The vertex ${\bar{v}}_1$ is the only one outside of C that is adjacent to $v_2$

• The vertex ${\bar{u}}_2$ is the only one outside of C that is adjacent to $u_1$ and not adjacent to $v_1$.

• The vertex ${\bar{v}}_2$ is the only one outside of C that is adjacent to $v_1$ and not adjacent to $u_1$.

• The vertex ${\bar{b}}$ is the only one outside C both adjacent to $u_1$ and $v_1$

• A B-adjacent vertex is not adjacent to $u_1$ nor to $v_1$ but is adjacent to b

• Finally, a vertex from $V_{G \setminus H}$ which is not B-adjacent is neither adjacent to $u_1$, $v_1$ nor b.

Therefore properties $(p_h)$ and $(p_b)$ are satisfied by (H, B, C) which is a 1-layered gadget for f. $\square$

Fig. 5

The 2-local gadget. The set $C_r$ contains the red and $B_r$ contains the square vertices

Definition 13

(r-local gadget) Given a positive integer r, let $H_r$ be the bipartite planar graph of size $(2r+2)^2$ such that:

• Its vertices are denoted $u_i^j$ and $d_i^j$ for $i \in [\![2r+2]\!]$ and $j \in [\![r+1]\!]$.

• For $i \in [\![2r+2]\!]$ and $j \in [\![r+1]\!]$, we define $U_i = \{u_i^k \; | \; k \in [\![r+1]\!]\}$, $U^j = \{u_k^j \; | \; k \in [\![2r+2]\!]\}$, $D_i = \{d_i^k \; | \; k \in [\![r+1]\!]\}$ and $D^j = \{d_k^j \; | \; k \in [\![2r+2]\!]\}$. Furthermore, as the vertices $u^i_j$ (resp. $d^i_j$) will form a cycle, we consider in the following that if $i \not \in [\![2r+2]\!]$, $u_i^1 = u_{1+((i-1)\pmod {2r+2})}^1$ (resp. $d_i^1 = d_{1+((i-1)\pmod {2r+2}))}^1$).

• For each $i \in [\![2r+2]\!]$, the vertex $u_i^1$ is adjacent to $d_i^1$, the vertices of $U_i$ form a path such that $d(u_i^1,u_i^j) = j-1$ for each $j \in [\![r+1]\!]$ and the vertices of $D_i$ form a path such that $d(d_i^1,d_i^j) = j-1$ for each $j \in [\![r+1]\!]$.

• The vertices of $U^1$ form a cycle such that $u_i^1$ is adjacent to $u_{i+1}^1$ for each $i \in [\![2r+2]\!]$, and the vertices of $D^1$ form a cycle such that $d_i^1$ is adjacent to $d_{i+1}^1$ for each $i \in [\![2r+2]\!]$.

We define the sets $B_r = \{u_1^1, d_{2r+2}^1\}$ and $C_r = U^1 \cup D^1$.

The triple $(H_r,B_r,C_r)$ is called ther-local gadget (see Fig. 5).

Proposition 5

Letfbe anr-local distance identifying function. Then ther-local gadget is an (f, r)-local gadget.

Proof

By construction, for each $i \in [\![2r+2]\!]$, the r-neighborhood of $u_i^{r+1}$ is $U_i$. An r-local distance identifying set must contain a vertex of $U_i$ to r-dominate $u_i^{r+1}$. Symmetrically, it must contain a vertex of $D_i$ to r-dominate $d_i^{r+1}$. Therefore, $C_r$ has optimal size and $(p_s)$ is verified. The condition $(p_d)$ is trivially verified, and the condition $(p_l)$ is also easily verified because of the vertices $u_i^1$ for $i \in [\![r+1]\!]$.

We have to prove $(p_h)$ and $(p_b)$. We first focus on vertices of $H_r$. We repeatedly use the property $(\beta _1)$ of Definition 4 when two vertices present different intersections between $C_r$ and their respective r-neighborhood. For $j \in [\![r+1]\!]$, let $uU_r(j) = 2r+3-2j$ be the number of r-neighbors of $u_i^j$ (with $i \in [\![2r+2]\!]$) belonging to the line $U^1$ of the code. Similarly, let $dU_r(j) = max(0,uU_r(j)-2)$ be the number of r-neighbors of $d_i^j$ belonging to $U^1$, let $uD_r(j) = dU_r(j)$ be the number of r-neighbors of $u_i^j$ belonging to $D^1$ and let $dD_r(j) = uU_r(j)$ be the number of r-neighbors of $d_i^j$ belonging to $D^1$.

As $uU_r > uD_r$ while $dU_r < dD_r$, $C_r$ distinguishes $u_i^j$ and $d_{i'}^{j'}$ for any relevant $i,j,i'$ and $j'$. Now, as the graph is symmetrical, we only have to consider vertices of form $u_i^j$.

For integers j and $j'$ such that $0< j< j' < r+2$, we have $uU_r(j) > uU_r(j')$, and therefore $C_r$ distinguishes $u_i^j$ and $u_{i'}^{j'}$, for any relevant $i,i',j$ and $j'$ such that $j \ne j'$.

It remains to distinguish $u_i^j$ and $u_{i'}^j$ for $i \ne i'$. A vertex $u_i^j$ is adjacent to every $u_k^1$ such that $i-r-1+j \le k \le i+r+1-j$, an interval of size $uU_r(j) < 2r+2$. We deduce that the r-neighbors of $u_i^j$ belonging to $U^1$ and the r-neighbors of $u_{i'}^j$ belonging to $U^1$ are different since they can be represented by two non-empty and non-covering intervals shifted by $i-i'$ (and $0< i-i' < 2r+2$).

We now have to consider the $B_r$-extensions of $H_r$. Let the graph G of vertex set $V_G$ be a $B_r$-extension of the graph $H_r$ of vertex set $V_H$, and let $N_B$ be the set of $B_r$-adjacent vertices. Note that $u^1_1$ was already at distance 2 of $d^1_{2r+2}$ so the $B_r$-adjacent vertices do not introduce any short cut. To distinguish $u \in V_H$ and $v \in V_{G \setminus H}$, it is sufficient to observe that by symmetry of the graph $H_r$, the number of r-neighbors of v belonging to $U^1$ is equal to the number of r-neighbors of v belonging to $D^1$, which is not the case for u (recall functions $uU_r$ and $uD_r$, and functions $dU_r$ and $dD_r$). Finally, to distinguish $u \in N_B$ and $v \in V_{G \setminus H} \setminus N_B$, we just have to use the guaranteed vertex $c \in C_r$ such that $d(c,B_r) = r-1$ and property $(\beta _1)$. This concludes the proof of the conditions $(p_h)$ and $(p_b)$.

We lastly observe that because $H_r$ is a bipartite planar graph, the r-local gadget is planar since $|B_r| = 2$, and it is bipartite since $d(u^1_1,d^1_{2r+2}) = 2$. $\square$

With Propositions 4 to 5, we can now prove the Theorems 4 to 6.

Proof

(Theorems4and6for each 1-layeredidentifying functionfand$r \in [\![\infty ]\!]$).

We first suggest a reduction from PlanarHitting Set to (f, r)-DIS based on the bipartite planar 1-layered gadget (H, B, C). Let $(\varOmega ,{\mathcal {S}})$ be an instance of PlanarHitting Set with $|\varOmega | = n$ and $|{\mathcal {S}}| = m$ such that $m = {\mathcal {O}}(n)$. According to Proposition 2, the bipartite apex graph $G = \varPhi ^*[H,B](\varOmega ,{\mathcal {S}})$ has size $n'$ linear in $n+m = {\mathcal {O}}(n)$ and may be built in polynomial-time in its size. Recall that (H, B, C) is an (f, r)-gadget by Proposition 4. By Theorem 7, G admits an (f, r)-distance identifying set of size $k' = k+|C|(n+m)$ if and only if ${\mathcal {S}}$ admits a hitting set of size k. Thus, an algorithm solving (f, r)-DIS in $2^{o(\sqrt{n'})}$ would solve PlanarHitting Set in time $2^{o(\sqrt{n})}$, a contradiction to Theorem 3 (assuming ETH).

We adapt the previous argumentation to get a reduction from Hitting Set to (f, r)-DIS, the instance $(\varOmega ,{\mathcal {S}})$ belonging now to the Hitting Set problem. According to Proposition 3, the bipartite graph $G = \varPsi [H,B,1](\varOmega ,{\mathcal {S}})$ has size $n'$ linear in $n+m = {\mathcal {O}}(n)$ and may also be built in polynomial-time in its size. By Theorem 8, G admits an (f, r)-distance identifying set of size $k' = k+|C|(b_{n+1}+b_{m})$ if and only if ${\mathcal {S}}$ admits a hitting set of size k. Thus, an algorithm solving (f, r)-DIS in $2^{o(n')}$ would solve Hitting Set in time $2^{o(n)}$, contradicting Theorem 2 when assuming ETH. Moreover, a parameterized algorithm solving (f, r)-DIS in $2^{{\mathcal {O}}(k)} \cdot {n'}^{{\mathcal {O}}(1)}$ would be in contradiction with Theorem 1 when assuming ${\mathsf{W}[2]} \ne \mathsf{FPT}{}$. $\square$

Proof

(Theorems5and6for eachr-local identifying functionf) First, we suggest a reduction from PlanarHitting Set to (f, r)-DIS based on the bipartite planar local (f, r)-gadget (H, B, C). Let $(\varOmega ,{\mathcal {S}})$ be an instance of PlanarHitting Set with $|\varOmega | = n$ and $|{\mathcal {S}}| = m$ such that $m = {\mathcal {O}}(n)$. According to Proposition 2, the (bipartite) planar graph $G = \varPhi [H,B,r](\varOmega ,{\mathcal {S}})$ has size $n'$ linear in $n+m = {\mathcal {O}}(n)$ and may be built in polynomial-time in its size. By Theorem 7, G admits an (f, r)-distance identifying set of size $k' = k+|C|(n+m)$ if and only if ${\mathcal {S}}$ admits a hitting set of size k. Thus, an algorithm solving (f, r)-DIS in $2^{o(\sqrt{n'})}$ would solve PlanarHitting Set in time $2^{o(\sqrt{n})}$, a contradiction to Theorem 3 (assuming ETH).

We adapt the previous argumentation to get a reduction from Hitting Set to (f, r)-DIS, the instance $(\varOmega ,{\mathcal {S}})$ belonging now to the Hitting Set problem. In this case, we only have to assume the existence of a (bipartite) local (f, r)-gadget (H, B, C). According to Proposition 3, the bipartite graph $G = \varPsi [H,B,r](\varOmega ,{\mathcal {S}})$ has size $n'$ linear in $n+m = {\mathcal {O}}(n)$ and may also be built in polynomial-time in its size. By Theorem 8, G admits an (f, r)-distance identifying set of size $k' = k+|C|(b_{n+1}+b_{m})$ if and only if ${\mathcal {S}}$ admits a hitting set of size k. Thus, an algorithm solving (f, r)-DIS in $2^{o(n')}$ would solve Hitting Set in time $2^{o(n)}$, contradicting Theorem 2 when assuming ETH. Moreover, a parameterized algorithm solving (f, r)-DIS in $2^{{\mathcal {O}}(k)} \cdot {n'}^{{\mathcal {O}}(1)}$ would be in contradiction with Theorem 1 when assuming ${\mathsf{W}[2]} \ne \mathsf{FPT}{}$. $\square$

5 Conclusion

In this paper, we showed generic tools to analyze identifying problems and their computational lower bounds. Questions about upper bounds naturally arise. While the general $2^{o(n)}$ lower bound under ETH is clearly tight, because the naive algorithm consisting in enumerating all the subsets of vertices has a complexity of $2^{O(n)}$, it is unknown whether the planar/apex $2^{o(\sqrt{n})}$ lower bound is tight. Furthermore, there is still a gap between the computational lower bound provided by Theorem 6 and the elementary upper bound from Proposition 1 in the local cases. We wonder if local problems may be solved in $k^{{\mathcal {O}}(k)} \cdot n^{{\mathcal {O}}(1)}$. Notice that a polynomial kernel would imply such a complexity (but the converse is not true). For non-local problems, an FPT upper bound is globally unknown. In particular, ${\mathsf{W}[2]}$-hard problems like MD cannot admit FPT algorithms unless ${\mathsf{W}[2]} = \mathsf{FPT}{}$. Then, which non-local problem is ${\mathsf{W}[2]}$-hard? We mention that we actually get an FPT reduction from Hitting Set to some scarce non-local problems (however including MD) proving their ${\mathsf{W}[2]}$-hardness, but the family of involved problems is not precise nor wide. Nevertheless, we remark that most of our reductions may be generalized to the oriented version of Distance Identifying Set sometimes even for strongly connected graphs –this is due to the fact that the paths in our distance identifying graphs and gadgets may often be seen as oriented–.