On the Distance Identifying Set meta-problem and applications to the complexity of identifying problems on graphs

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 mainly focus on two families of problem from the meta-problem: the first one, called bipartite gifted 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: - the 1-layered problems are NP-hard even in bipartite apex graphs, - the bipartite gifted local problems are NP-hard even in bipartite planar graphs, - 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, - even restricted to bipartite graphs, they do not admit parameterized algorithms in $2^{O(k)}.n^{O(1)}$ except if W[0] = W[2]. 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 in 2013.

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 mainly focus on two families of problem from the meta-problem: the first one, called bipartite gifted 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: • the 1-layered problems are NP-hard even in bipartite apex graphs, • the bipartite gifted local problems are NP-hard even in bipartite planar graphs, • assuming ETH, all these problems cannot be solved in 2 o( √ n) when restricted to bipartite planar or apex graph, respectively, and they cannot be solved in 2 o(n) on bipartite graphs, • even restricted to bipartite graphs, they do not admit parameterized algorithms in 2 O(k) · n O(1) except if W[0] = W [2]. 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 in [19].

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 [22,28], graph isomorphism [2] or logical definability of graphs [23]. The versions of these problems in hypergraphs have been studied under different names in [5], [6] and [7].
Given a graph G with vertex set V , the classical identifying sets are defined as follows: • IC: Introduced by Karposky et al. [22], a set C of vertices of G is said to be an identifying code if none of the sets N [v] ∩ C are empty, for v ∈ V and they are all distinct.
• LD: Introduced by Slater [25,26], a set C of vertices of G is said to be a locating-dominating set if none of the sets N [v] ∩ C are empty, for v ∈ V \ C and they are all distinct. When not considering the dominating property (N [v] ∩ C may be empty), these sets have been studied in [2] as distinguing sets and in [23] as sieves.
• MD: Introduced independently by Harary et al. [17] and Slater [24], 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) = 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 [9,10]. Charon et al. showed the NP-hardness when restricted to bipartite graphs [8], while Auger showed it for planar graphs with arbitrarily large girth [1]. For trees, there exists a linear algorithm [25].
Metric Dimension is also NP-hard, even when restricted to Gabriel unit disk graphs [16,20]. Epstein et al. [13] 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. Additionally, Diaz et al. [11] proved a quite tight separation: the problem is polynomial on outerplanar graphs whereas it remains NP-hard on bounded degree planar graphs.
In a recent publication, 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 [12] 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 W [2] = FPT (= W[0]). Indeed, Hartung et al. [18,19] showed MD is W[2]-hard for bipartite subcubic graphs. The problem is however FPT on families of graphs with degree ∆ growing with the number of vertices because the size k of a resolving set must satisfy log 3 (∆) < k. Finally, Foucaud et al. [15] provided a FPT algorithm on interval graphs.  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 set/element-gadget technique, the noteworthy adaptation of the clause/variable-gadget technique from SAT to Hitting Set.
As we provide a 1-layered generic gadget, the 1-layered reductions operate without condition. For local problems, the existence of a local gadget is not always guaranteed. Thus, a local reduction operates only if a local gadget is provided. 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.
The reductions in general graphs are designed to exploit the W[2]-hardness of Hitting Set parameterized by the solution size k HS of an hitting set, hereby using: Theorem 1 (folklore). Let n HS and m HS be the number of elements and sets of an Hitting Set instance, and k HS be its solution size. A parameterized problem with parameter k admitting a reduction from Hitting Set verifying k = O(k HS +log(n HS +m HS )) does not have a parameterized algorithm running in Proof. Given a reduction from Hitting Set to a parameterized problem Π such that the reduced parameter satisfies k = O(k HS + log(n HS + m HS )) and the size of the reduced instance verifies n = (n HS + m HS ) O(1) , an algorithm for Π of running time 2 O(k) · n O(1) is actually an algorithm for Hitting Set of running time 2 O(k HS ) · (n HS + m HS ) O(1) , meaning that Hitting Set is FPT, a contradiction to its W[2]-hardness (otherwise W[2] = FPT).
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. From the best of our knowledge, this merging gadgets technique has never been employed.
The organization of the paper is as follows. After a short reminder of the computational properties of Hitting Set, Section 2 contains the definitions of distance identifying functions and sets, allowing us to precise the computation lower bounds we obtain. The Section 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 Section 4 as well as the proofs of the main theorems.
2 Definition of the Meta-Problem and Related Concepts

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 [[∞]] = N >0 ∪ {∞}. 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 of u is denoted by N (u), its closed neighborhood is , that is the set of vertices at distance less than r + 1 of u. For r = ∞, the ∞-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] ∩ D is non-empty. Thus an ∞-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 ∈ X, y ∈ 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 ∆ Y , and the 2-combination of a set X is denoted P 2 (X) Given is present in E if and only if the element u i belongs to the subset S j . Henceforth, a hitting set of S is equivalent to a subset C of V Ω that dominates V S . We call φ(Ω, S) the associated graph of (Ω, S).

Planar Hitting Set
Input: An instance (Ω, S) of Hitting Set such that φ(Ω, S) is planar. Output: A hitting set C of S of minimal size. We also consider the parameterized version Planar Hitting Set(k) of the latter problem. Proof. Let Φ be the set of the n variables present in the set C of clauses of an instance of SAT. For each variable ϕ of Φ, we add two fresh elements u ϕ andū ϕ to the universe Ω Φ representing the two possible affectations of variable ϕ, and we create a set S ϕ = {u ϕ ,ū ϕ } that we append to the set S C of subsets of Ω Φ . The independence of the sets S ϕ implies that the existence 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 Φ. Finally, to determine if an affectation satisfies the set of clauses C, for each clause c ∈ C we append to S C the set of elements representing each literal present in the clause c. The equivalence between the satisfiability of C and the existence of a hitting set of S C of size n is immediate by construction. It remains to guarantee the planarity of the associated graph φ(Ω Φ , S C ) . To do so, we actually apply the reduction on a restriction of SAT named Separate Simple Planar SAT (See [27] for a precise definition). Adding the sparsifying lemma from [21], 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(  • r-MD: a subset C of V is a 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 ∈ N r [u] ∪ N r [v] and d(u, w) = d(v, w).
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: We need to require some useful properties on identifying functions to produce generic results. By mimicking the classical identifying sets, the main property we consider is that a vertex cannot distinguish two vertices at the same distance from it. Then: Definition 2 (distance function). A distance identifying function f is an identifying function such that for every graph G and all vertices u,v and w of G with u = v: Besides this mandatory criterion, we suggest two paradigms related to the neighborhood of a vertex.
]. 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 to 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: , an i-local identifying function f is an identifying function such that for every graph G and all vertices u, v, w of G with u = v: , 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 = v: In the following, given an identifying function f and three vertices u, v, w of a graph G, we say that w f -distinguishes u and v if and We are now ready to define the Distance Identifying Set meta-problem. Given a distance identifying function f and r ∈ [[∞]] as inputs of the meta-problem, the resulting problem is called (f, r)-Distance Identifying Set and denoted (f, r)-DIS. The 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. A problem is local if it is i-local for an integer i. Our local reductions will need a local gadget to operate: the subfamilies of local problems admitting a (bipartite) local gadget is called (bipartite) gifted local. We do not need to define gifted 1-layered as every 1-layered problem admits a 1-layered gadget. We also consider the parameterized version Distance Identifying Set(k).

Detailed Computational Lower Bounds
Using the Distance Identifying Set meta-problem, we get the following lower bounds:   Theorem 7. Let f, g and h be distance identifying functions such that f is 1-layered, g is q-local 0-layered and h is p-local and admits a local (bipartite) gadget. Let r ∈ [[∞]]. The (f, r)-, (g, q)and (h, p)-DIS problems are NP-hard, and do not admit: The parameter k denotes here the solution size of a relevant distance identifying set. All bounds still hold in the bipartite case (whenever the gadget associated with h is bipartite).
As a side result, the 1-layered general reduction answers a question of Hartung in [19]: Finally, notice that the parameterized lower bound from Theorem 7 may be complemented by an elementary upper bound inspired from the kernel of IC and LD of size 2 k + k: Proposition 8. For every r-local distance identifying function f , the (f, r)-Distance Identifying Set problem has a kernel of size (r + 1) k + k where k is the solution size. Therefore, it admits a naive parameterized algorithm running in O(n k+3 ) ∈ O * (r (k 2 ) ) time.
Proof. The kernel size simply relies on the fact that k vertices may characterize at most (r + 1) k r-neighbors using distances, while the parameterized algorithm just enumerates the n k set of k vertices of the input graph, trying them in O(n 3 ).
The proofs of the Theorems 4 to 7 will be given in Section 4.

The Distance Identifying Graphs
Consider the associated graph φ(Ω, S) as defined in Section 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 Ω i does not distinguish a vertex v 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 Ω i formally has to distinguish a vertex v 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 ∈ S, the set V S may contain two vertices v S j andv S j . Then, the role of v Ω i would be to distinguish them if and only if u i ∈ S j . To ensure that the vertex v Ω i distinguishes v S j and v S j when u i ∈ S j , we may use the properties (β 1 ) and (γ) of Definition 3 and 4 for the r-local and 1-layered problems, respectively. Precisely, when of Definitions 2 and 3. Hence, when 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 Ω and the vertices of a distance identifying set included in V Ω . However, it is clearly not sufficient since we also have to distinguish the couples of vertices of V Ω 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: The B-extensions of H such that V G\H contains exactly a B-adjacent vertex or two B-adjacent vertices but not connected to each other are called the B-single-extension and the B-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 Ω i , while a B-twin-extension contains a gadget with its two related vertices v S j andv S j . Piecing all together, we may adapt the associated graphs to the meta-problem: • finally, for each S j ∈ S and each u i ∈ S j , v Ω i is connected to v S j by a path of r − 1 vertices 1]]. When the problem is not local, we prefer the following identifying graph:  a (H, B, 1)-distance identifying graph with an additional vertex a called apex such that: H Ω i ) denotes the copy of B (resp. H).
• for each S j ∈ S, the apex a is adjacent to v S j andv S j . See Figure 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).
H Finally, all the other items of the proposition are direct by construction.
Having defined the (apex) distance identifying graphs, the main effort to obtain generic reduction from Planar Hitting Set is done. We now define relevant gadgets: Definition 9 ((f, r)-gadgets). Let f be a distance identifying function and r ∈ [[∞]]. Let H = (V H , E H ) be a connected graph, and B, C be two subsets of V H . We said that the triple (H, B, C) is a (f, r)-gadget if for every B-extension G of H:  Proof. We start by focusing on the equivalence between the first and second items.
Suppose first that P is a hitting set of (Ω, S) of size k. By denoting C Ω i and C S j the copies of C associated to the copies H Ω i and H S j of H, we suggest the following set I of size k + |C|(n + m) as a (f, r)-distance identifying set of G = Φ * [H, B](Ω, S):

Recall that by construction, G is a B
. This directly implies that I is an r-dominating set of G. Indeed, the condition (p d ) of Definition 9 implies that C The remaining apex is also r-dominated by any C Ω i , as it is We now have to show that I f -distinguishes G. We begin with the vertices of the gadget copies because the condition (p h ) implies that C Ω i ⊆ I f -distinguishes the vertices of H Ω i and G for every i ∈ [[n]], and I f -distinguishes the vertices of H S j and G for every j ∈ [[m]]. Thereby, we only have to study the vertices of the form v Ω i , v S j ,v 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 ∈ [[n]], we have: Enumerating the relevant i and i , we deduce that every couple of vertices is distinguished except when they are both of the form v S j orv S j for j, j ∈ [[m]]. But we may distinguish v S j orv S j for distinct j, j by applying ( . We now use the fact that P is a hitting set for (Ω, S). By definition of a hitting set, for any set S j ∈ S, there exists a vertex u i ∈ P such that u i ∈ S j . We observe that In the other direction, assume that I is a distance identifying set of G of size k + |C|(n + m). As every set of S is not empty, we may define a function ϕ : We suggest the following set P as an hitting set of S of size at most k: We claim that the only vertices that may f -distinguish v S j andv S j are themselves and the vertices v Ω i such that u i ∈ S j . To prove so, we apply propriety (α) of Definition 2: • both v S j andv S j are B S j -adjacent, so they are at the same distance of any vertex of H S j . We deduce that v S j andv S j are f -distinguished only if either one on them belongs to I (in that case u ϕ(j) ∈ P ∩ S j ) or there exists v Ω i ∈ I such that u i ∈ S j (and then u i ∈ P ∩ S j ). It remains to show that |P | ≤ k. By the condition (p s ) of Definition 9, we know that ]. In the first direction, suppose that P is a hitting set of (Ω, S) of size k, the (g, r)-distance identifying set I of G = Φ[H, B, r](Ω, S) is defined identically as in the equivalence of the first and second items of the current theorem: Using conditions (p d ) and (p l ) of Definitions 9 and 10, I is clearly an r-dominating set of G. Indeed, by (p d ) every vertex belonging to a copy of the gadget is r-dominated. Additionally, every vertex outside of the copies of the gadgets is at distance at most r of a copy by construction, but there exists a vertex b ∈ B ∩ C (so a relevant copy in I) by (p l ).
To prove that I f -distinguishes G, the strategy is differing from the previous equivalence only on the l k i,j vertices and when distinguishing v S j andv S j as we will see. Recall that by construction, G is a ]. Thereby, we only have to study the vertices of the form v Ω i , v S j ,v S j , and the vertices l k i,j . To distinguish them, we mainly use the condition (p b ). We observe that for each distinct i, i ∈ [[n]] (they exist as n > 1) : It remains to distinguish v S j andv S j for j, j ∈ [[m]]. If j and j are distinct we may use (p b ) on the copy H S j of the gadget H. We may assume j = j . We now use the fact that P is a hitting set for (Ω, S). By definition of an hitting set, for any set S j ∈ S, there exists a vertex u i ∈ P such that u i ∈ S j . We observe that 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 k i,j . Let define 1]] and u i ∈ S j }. As every set of S is not empty, we may define a function ϕ : ] such that u ϕ(j) ∈ S j . We suggest the following set P as an hitting set of S of size at most k: ], let us show that the only vertices that may f -distinguish the couple (v S j ,v S j ) are themselves and the vertices from L i (and not only v Ω i ) such that u i ∈ S j . Every vertex from H S j is at the same distance to v S j andv S j and thus cannot f -distinguishes them because of the distance property (α). Every vertex not in H S j , not in L i for every i ∈ [[n]] such that u i ∈ S j and different from v S j andv S j is at distance at least r +1 of the two latter vertices. Thus, because of the propriety (β 2 ) of Definition 3 (a vertex cannot distinguish two vertices outside of its r-neighbourhood) any of these vertices does not f -distinguish v S j andv S j . We deduce that v S j andv S j are f -distinguished if and only if either one of them belongs to I (in that case u ϕ(j) ∈ P ∩ S j ) or there exists i ∈ [[n]] such that u i ∈ S j and I ∩ L i = ∅.
The proof that |P | ≤ k is provided by (p s ), we know that Obviously, the second and third items are equivalent since they are both equivalent to the first item, which concludes the proof.

Binary Compression of Gadgets
The Theorem 10 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 + log 2 (n) and introduce a new graph:  • for each j ∈ [[m]], we add two non-adjacent vertices v S j andv S j . They are B S k -adjacent for each k ∈ [[b m ]] such that the k th bit of the binary representation of j is 1.
]] such that the k th bit of the binary representation of i is 1.
• for each S j ∈ S and each u i ∈ S j , we add the edge (l r−1 i , v S j ). 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, we have (b n+1 + b m )|H| + n + 2m + n(r − 1) + r = |H|(b n+1 + b m ) + r(n + 1) + 2m Finally, the two last items of the proposition are also direct by construction.
Theorem 12. Let (Ω, S) be an instance of Hitting Set such that |Ω| = n, |S| = m. Let (H, B, C) be a (f, r)-gadget for a 1-layered identifying function f and let (H , B , C ) be a local (g, q)-gadget. The following propositions are equivalent: • there exists a hitting set of S of size k.  Proof. Suppose again that P is a hitting set of (Ω, S) of size k. By denoting C Ω i and C S j the copy of C (respectively C ) associated to the copy H Ω i and H S j of H (respectively H ), we suggest the following set I of size k + |C|(b n+1 + b m ) as a (f, r)-distance identifying set of G = Ψ[H, B, 1](Ω, S) (respectively (g, q)-distance identifying set of G = Ψ[H , B , q](Ω, S)): ). This directly implies that I is an r-dominating set of G (respectively q-dominating set G ).
We only have to show that I f -distinguishes G (respectively 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 S j ,v 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 S j or v S j for j ∈ [[m]] (or if they are both of the form a k or l k i for k ∈ [[q − 1]] for G , G not containing such vertices).
To distinguish v S j andv S j for j ∈ [[m]]. We now use the fact that P is a hitting set for (Ω, S). By definition of an hitting set, for any set S j ∈ S, there exists a vertex u i ∈ P such that u i ∈ S j . We observe that by construction of G ) and that v Ω i ∈ I by definition of I. Since f is 1-layered (respectively g is q-local), I f -distinguishes and g-distinguishes v S j andv S j . For G , it remains to distinguish a k and l k i for k ∈ [[q − 1]] and i ∈ [[n]]. We recall that in a (g, q)-local gadget (H , B , C ), there exists c ∈ C such that d(c, B ) = k − 1 for each k ∈ [[q]]. Then we may use characteristic function of the power set together with property (β 1 ) of a q-local function to distinguish them.
In the other direction, assume that I is a (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 S is not empty, we may define a function ϕ : We suggest the following set P as an hitting set of S of size at most k: 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 S j ,v S j ) are themselves and the vertices of form l k i such that u i ∈ S j . To prove so, we apply propriety (α) from Definition 2 (a vertex cannot distinguish two vertices at the same distance from it) on the following enumeration on G: Figure 4: The 1-layered gadget (H, B, C). C contains the colored vertices.
because of the path formed by the vertices of form a k . The enumeration on G is identical when replacing r by q. We deduce that v S j andv S j are fdistinguished if and only if either one on them belongs to I (in that case u ϕ(j) ∈ P ∩ S j ) or if there exists l k i ∈ I such that u i ∈ S j and k + 1 ∈ [[r]] (and then u i ∈ P ∩ S j ).

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 Section 3. The existence of the gadgets rely on the following tool lemma: Lemma 2 (Twins Lemma). Let x and y be two vertices of a graph G such that N (x) = N (y). Then any distance identifying set of G contains either x or y.
Proof. Because N (x) = N (y), for every vertex u of G, if u ∈ {x, y}, then d(u, x) = d(u, y). Thus, by property (α) of a distance identifying set, u may distinguishes x and y if and only if u ∈ {x, y}, implying that a distance identifying set must contain either x or y.
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,b, • The vertices u 1 , u 2 ,ū 1 andū 2 form a cycle as well as the vertices v 1 , v 2 ,v 1 andv 2 .
We define the sets B = {b,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).  We define the sets The triple (H r , B r , C r ) is called the r-local 0-layered gadget (see Fig. 5).
For each positive integer r, r-LD and r-MD are r-local 0-layered problems, whereas r-IC is not 0-layered. We define a specific gadget for this remaining problem. ]. We also denote a 0 as a 1 r+1 and a 2 r+1 and we denote b 0 as b 1 r+1 , b 2 r+1 , a 1 0 and a 2 0 .
• the edges are all included in the six following paths from a 0 to a r+1 such that d(a 0 , a i ) = i for i ∈ [[r + 1]]. - from a 1 0 to a 1 r+1 such that d(a 1 0 , The triple (H r , B r , C r ) is called the r-IC gadget (see Fig. 6).
As expected, we have the following propositions: Let i and j be two integers such that 0 ≤ i < j ≤ r, and suppose that we have two vertices x and y such that x is either b 1 j or b 2 j , and y is either b 1 i or b 2 i . Then a 1 j−1 distinguishes x and y because . The same reasoning holds when x ∈ {a 1 j , a 2 j } and y ∈ {a 1 i , a 2 i }. Furthermore, we have N r [b r+1 ] ∩ C r = {b r+1 } and there are no other vertices x such that N r [x] ∩ C r = {b r+1 }. So b r+1 is distinguished from any other vertices of G. The same reasoning proves that a r+1 is distinguished from any other vertices too.
The vertex a 0 distinguishes any vertex of , one can notice that b 1 r (resp. a 1 r ) distinguishes x and b 0 .
and so x and y are distinguished. Otherwise, if y is b r+1 (resp. a r+1 ), then b r+1 (resp. a r+1 ) distinguishes x and y. Otherwise, a 0 distinguishes the two vertices. Then (H r , B r , C r ) verifies property (p h ).
Let us now prove that H r distinguishes N Br and V G\Hr \ N Br . Let x be a vertex of the former and y be a vertex of the latter. i and a 2 i implies that either a 1 r−i or a 2 r−i must be in S. Furthermore, as N r [b r+1 ] ∩ S (resp. N r [a r+1 ]∩S) cannot be empty by definition of an identifying code, then there exists i ∈ [r +1] such that b i ∈ S (resp. a i ∈ S). We conclude that there are at least 2r +4 vertices of H r (that is the size of C r ) in S, proving that property (p s ) holds. This proves that (H r , B r , C r ) is a (f, r)-gadget. By construction, we can easily see that (H r , B r , C r ) verifies the property of local gadgets. Therefore, this is also a local-gadget.
With Propositions 13 to 15, we can now prove the Theorems 4 to 7. Proof of Theorems 6 and 7 for each r-local 0-layered identifying function f . By Proposition 14, the r-local 0-layered gadget is a local bipartite planar (f, r)-gadget for every r-local 0-local distance identifying function f . Then, Theorems 5 and 7 for r-local identifying function f apply and directly yield the current theorems.

Conclusion
In this paper, we showed generic tools to analysis identifying problems and their computational lower bounds. This study opens some new questions. First of all, we observe that our toolbox does not contain a r-local gadget. Does one exist? Furthermore, there is still a gap between the computational lower bound provided by Theorem 7 and the elementary upper bound from Proposition 8 in the local cases. We wonder if local problems may be solved in k O(k) · n O(1) . Notice that a polynomial kernel would imply such a complexity (but the reciprocal is not true). For non-local problems, an FPT upper bound is globally unknown. In particular, W[2]-hard problems like MD cannot admit FPT algorithms unless W[2] = FPT. Then, which non-local problem is W[2]-hard? We mention that we actually get a FPT reduction from Hitting Set to some scarce non-local problems (however including MD) proving their 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 the 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-. Thus, we inform the community that the oriented version of MD (studied for Cayley graphs in [14]) remains W[2]-hard.