Locating-dominating sets: from graphs to oriented graphs

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locating-dominating set of a directed graph D is a subset S of its vertices such that two vertices not in S have distinct and non-empty in-neighbourhoods in S. The directed location-domination number of D, denoted by γ LD (D), is the size of a smallest locating-dominating set of D.
Two oriented graphs with the same underlying graph can have a very different behaviour towards locating-dominating sets. Let us illustrate it on tournaments that are oriented complete graphs. Transitive tournaments (i.e. acyclic tournaments) have directed location-domination number ⌈n/2⌉ whereas one can construct locating-dominating sets of size ⌈log n⌉ for a well-chosen orientation of K n [28]. Following the idea of Caro and Henning for domination [6] and the work started by Skaggs [28], we study in this paper the best and worst orientations of a graph for locating-dominating sets. A similar line of work has been recently initiated for the related concepts of identifying codes [9] and metric dimension [2].
The two parameters that will be considered in this paper are the following. The lower directed location-domination number of an undirected graph G, denoted by

Outline of the paper
Basic definitions, some background and first results are given in Section 2. Section 3 is dedicated to the study of the best orientations whereas Section 4 focuses on the worst orientations.
Main results on best orientations. We first give basic results on → γ LD (G) and relations with classical parameters of graphs. Skaggs [28] proved in 2007 that for any graph G, → γ LD (G) ≤ γ LD (G). We refine this inequality by proving that, in graphs without cycles of size 4 (as a subgraph), → γ LD (G) and γ LD (G) coincide. As a consequence, computing → γ LD (G) is NP-complete. Two vertices are twins if they have the same open or closed neighbourhood. Twins play an important role in locating-dominating sets since any locating-dominating set must contain at least one vertex of each pair of twins. As a consequence, if G is a star on n vertices, then → γ LD (G) = n−1. In Section 3.3, we prove that this function can be drastically improved when the graph G is twinfree, which is one of the main contributions of our paper. Theorem 1. Let G be a twin-free graph of order n with no isolated vertices, then → γ LD (G) ≤ n/2.
The fact that any twin-free graph of order n satisfies γ LD (G) ≤ n/2 is a notorious conjecture, left open in [12,16] for instance. Conjecture 2 ( [16]). If G is a twin-free graph of order n, then γ LD (G) ≤ n/2.
The proof of Theorem 1 holds in two steps. First, we show in Section 3.2 that → γ LD (G) is the smallest undirected location-domination number among all the (connected) spanning subgraphs of G. Then we prove in Section 3.3 that there exists a spanning subgraph for which the condition is satisfied. In particular, our result implies a weakening of Conjecture 2 since we prove that any twin-free connected graph G on n vertices admits a spanning subgraph H with γ LD (H) ≤ n/2.
We then focus on (almost) regular graphs in Section 3.4 and prove, using a probabilistic argument, that there exists a constant c d such that, if G is d-regular, We continue this subsection by giving some bounds using independence and matching numbers.
Main results on worst orientations. In Section 4.1, we give some examples and relate → Γ LD (G) with some classical graph parameters. In particular, we prove that → Γ LD (G) ≥ γ LD (G) if G does not have any cycle of length 4 as a (not necessarily induced) subgraph. Moreover we prove that if G is a C 4 -free bipartite graph (which in particular, contains the class of trees), then → Γ LD (G) = α(G) where α(G) is the maximum size of an independent set.
In Section 4.2, we prove that → Γ LD (G) ≥ γ LD (G) is satisfied for other graph classes such as bipartite graphs, cubic graphs, and outerplanar graphs. Somehow surprisingly at first glance, → Γ LD (G) ≥ γ LD (G) is not always true. In [13], Foucaud et al. have shown that for a complete graph K n on n vertices we have → Γ LD (K n ) = ⌈n/2⌉ but γ LD (K n ) = n − 1. We prove that the existence of twins is not the reason why this inequality fails since we exhibit a family of twin-free graphs for which the ratio → Γ LD (G)/γ LD (G) tends to 1/2. We did not succeed to bound this ratio by a constant. However, we prove that → Γ LD (G) ≥ γ LD (G)/⌈log 2 (∆(G))+1⌉. We left the existence of a constant bounding → Γ LD (G)/γ LD (G) as an open problem. Finally, in Section 4.3, we provide some lower bounds on → Γ LD (G) using the number of vertices. For numerous classes of graphs, we actually have → Γ LD (G) ≥ c 1 · n c2 where c 1 and c 2 are constant. This is true for perfect graphs (with c 2 = 1/2), C 3 -free graphs, claw-free graphs and actually for any χ-bounded class of graphs with a polynomial χ-bounding function. However, we left as an open problem the existence of a graph G on n vertices such that → Γ LD (G) is logarithmic on n. Note that we did not find the complexity of computing → Γ LD (G). In particular, it is not clear that this problem belongs to NP.

Notations
We give in this subsection the main definitions and notations we are using along the paper. The reader may refer to some classical graph theory books like [4] for missing definitions.
Let G = (V, E) be an undirected and simple graph. We will generally denote by n the number of vertices of G. We denote by N G (u) (or N (u) when G is clear from context) the open neighbourhood of a vertex u, that is the set of neighbours of u. And we denote by . The degree of a vertex u, denoted by d(u), is the size of N (u). The minimum and maximum degree of G are respectively denoted by δ(G) and ∆(G). A leaf is a vertex of degree 1. A The complete graph on n vertices is denoted by K n . The complete bipartite graph with size n and m is denoted by K n,m . A star is a graph isomorphic to K 1,m . The star with three leaves, K 1,3 , is also called a claw. The cycle on n vertices is denoted by C n whereas the path on n vertices is denoted by P n . The girth of a graph G is the length of a shortest cycle in G. If G does not contain any cycle we say that G has infinite girth. A set S of vertices is independent if they are pairwise non adjacent. A set S is an edge cover if every edge has at least one endpoint in S. A set of edges M is a matching if no two edges in M share an endpoint. In a graph G, we denote the cardinalities of maximum independent sets and matchings by α(G) and α ′ (G), respectively. Moreover, the cardinality of a minimum edge cover is denoted by β(G).
Let S be a subset of V . Set S is a dominating set of G if any vertex of G is either in S or adjacent to a vertex of S. The minimum size of a dominating set is denoted by γ(G). We denote by I G (S; u) (I(u) for short) the set N G [u] ∩ S that is the neighbours in S of a vertex u. Note that S is a locating-dominating set if for each vertex u, I(u) is non-empty (since S is a dominating set) and for each pair of distinct vertices u, v ∈ V \ S, we have I(u) = I(v). We say that a vertex s ∈ S separates u and v if s is in exactly one of sets I(u) and I(v). Note that any locating-dominating set must intersect any pair of twins. The minimum size of a locating-dominating set of G is denoted by γ LD (G).
These notions are similarly defined for directed graphs. In this paper, we mostly consider directed graphs derived from orienting an undirected graph. Let G = (V, E) be a simple undirected graph. An orientation of G is a directed graph D on V where every edge uv of G is ever oriented from v to u (resulting to the arc (u, v) in D) or from v to u (resulting to the arc (v, u)). In particular, all the directed graphs considered are oriented and simple: if (u, v) is an arc then (v, u) is not. The undirected graph G is called the underlying graph of D. A tournament is an orientation of a complete graph. The open out-neighbourhood and in-neighbourhood of a vertex u of D are denoted by N + D (u) and N − D (u) whereas the closed out-and in-neighbourhood are denoted by N + . The maximum out-and in-degree are denoted by ∆ + (G) and ∆ − (G). A source is a vertex with no in-neighbours. Locating dominating sets are defined similarly as in the undirected case by considering the in-neighbourhoods. We denote by I D (S; u) (or I(u) for short) the set N − D [u] ∩ S, that is, the in-neighbours of u that are in a set S of vertices. The set S is a locating-dominating set of D if all the sets I D (S; u) are non empty and distinct for u / ∈ S. The minimum size of a locating-dominating set of D, called the minimum directed location-domination number, is denoted by γ LD (D).
We finally recall the two main parameters that we are considering along this paper. The lower directed location-domination number of an undirected graph G, denoted by

Preliminary results and examples
Let D be a digraph and u be a non-isolated vertex of D. Then V (D)\ {u} is a locating-dominating set of D. In particular, for any directed graph containing at least one edge, Γ d (D) ≤ n − 1. In [13], the authors have characterized digraphs that reach this extremal value. This characterization will be useful for studying the extremal values of → γ LD (G) and → Γ LD (G). A directed star is a (non necessarily simple) directed graph such that the underlying graph is a star. A bi-directed clique is a directed graph that contains all the possible arcs between two vertices. 2. D is a directed star; 3. V (D) can be partitioned into three (possibly empty) sets S 1 , C and S 2 , where S 1 and S 2 are independent sets, C is a bi-directed clique, and the remaining arcs in D are all the possible arcs from S 1 to C ∪ S 2 and those from C to S 2 .
In particular, any orientation of a star has location-domination number n − 1. In [13], the authors also proved a tight upper bound for tournament: As a consequence, the upper directed location-domination number of complete graphs is known: Concerning the best orientation of a complete graph, Skaggs proved in his thesis [28] that one can obtain the best possible number for → γ LD (G). For that, choose k vertices of the complete graphs and then for each other vertex, it is possible to orient the arcs to make any in-neighbourhood to the k chosen vertices.

Best orientation
In this section we focus on the best orientation. We first give basic results and links with classical parameters. Then we give another definition of → γ LD (G) using spanning subgraphs and use this definition to that → γ LD (G) ≤ n/2 if G is twin-free. We finally improve this last result in the case of almost regular graphs.

Basics
Theorem 8. Let G be a graph of order n. Then Proof. Let us prove (2). Let G be a graph on n vertices and let M be a maximum matching of G. Let V M be a subset of vertices containing exactly one vertex from each edge of M and C M be the set of vertices which are not endpoints of edges in M . Let C = V M ∪ C M . Note that |C| = n − α ′ (G). Choose any orientation D ′ of G where the edges in M have their tails in C and all the other edges between V \ C and C are oriented from V \ C to C. Now, C is a locatingdominating set in D ′ since all the vertices of V \ C have exactly one in-neighbour in V M and all of them are pairwise distinct.
We will show that these bounds are tight in Corollary 9 and Theorem 10. Using Theorem 3, we now provide a characterization of graphs reaching the extremal value → γ LD (G) = n − 1. Otherwise, let D be an orientation of G. Since Γ d (D) ≤ n − 1 we must actually have Γ d (D) = n − 1. Since G is not at star, then D must have the structure of the third condition of Theorem 3.
Thus V (G) can be partitioned to sets S 1 , C and S 2 satisfying the third condition of Theorem 3. Since C is a bi-directed clique in Theorem 3, we have |C| ≤ 1 because D is an oriented graph. Assume first that |C| = 1. If S 1 or S 2 are empty, then G is a star. If both of them are not empty, then G contains a triangle and there is an orientation D ′ of G with an oriented cycle. Then by Theorem 3, Γ d (D ′ ) ≤ n − 2, a contradiction.
If C = ∅, then G is a star if either |S i | = 1 for i ∈ {1, 2} and disconnected if either is an emptyset. But if |S i | ≥ 2, then again G contains a cycle and an orientation with an oriented cycle which is against the conditions of Theorem 3. Hence, the claim follows.
Let us prove that if G is without C 4 as a (not necessarily induced) subgraph, then it is actually an equality.
Theorem 10. Let G be a graph without C 4 as a subgraph. Then Proof. To prove the claim, let us show that, any directed locating-dominating set S of an orientation D of a graph G is also a locating-dominating set for G. Let D be an arbitrary orientation of G and S be a locating-dominating set of D. First note that S is indeed a dominating set of G. Thus, if S is not locating-dominating in G, then there exist u, v ∈ S such that . But then u, c 1 , v and c 2 is a cycle on four vertices, a contradiction.
In particular, Theorem 10 means that → γ LD (T ) = γ LD (T ) for any tree T . Let us complete this warm-up part by proving that finding the value of Locating-Dominating-Set Instance: A graph G, an integer k. Question: Is it true that γ LD (G) ≤ k?
Theorem 11. Locating-Dominating-Set and Lower-Directed-LD-Number are NP-complete for planar graphs of maximum degree 5 without C 4 as a subgraph.
Proof. Both problems are in NP. For Lower-Directed-LD-Number, a polynomial certificate for → γ LD (G) ≤ k is an orientation D of G and a locating-dominating set of D of size at most k. By Theorem 10, both values are equal in the class of graphs without C 4 . Thus, we just prove the result for Locating-Dominating-Set. We will reduce it from Dominating-Set.

Dominating-Set
Instance: A graph G, an integer k. Question: Is it true that γ(G) ≤ k?
We use the reduction of Gravier et al. [18,Figure 7]. Consider an instance (G, k) of Dominating-Set. Let G △ be the graph obtained by adding to each vertex of the graph a pendant triangle (see Figure 1). Then it is proved in [18] that G has a dominating set of size k if and only if G △ has a locating-dominating set of size k + n (where n is the number of vertices of G). Indeed, each triangle must contain at least one of the new vertices in a locating-dominating set and if there is exactly one vertex in a triangle, the vertex of the original graph must be dominated in the original graph.
Dominating-Set has been proved to be NP-complete even for planar graphs of maximum degree 3 and girth at least 5 [33]. If G is planar of maximum degree 3 and girth at least 5, then G △ is planar, of maximum degree 5, and does not contain C 4 as a subgraph. This implies our result.

Relation to spanning subgraphs
In this section, we prove a simple but important lemma that links → γ LD (G) with optimal locatingdominating sets of spanning subgraphs of G. This result will be used to prove several important results all along the section, but we illustrate its interest by first giving several simple lower bounds on → γ LD (G). of G such that an edge e between S and V \ S is oriented away from the vertex in S if e ∈ E(H) and if e ∈ E(H), then we orient edge e towards the vertex in S. Other edges can be oriented in any way. Observe that we have I D (S; w) = I H (S; w) for each vertex w ∈ S and hence, S is locating-dominating in D. Thus, Let us then show that for any orientation D ′ of G, there exists a spanning subgraph if and only if either u ∈ S and the edge is oriented away from u in D ′ or v ∈ S and the edge is oriented and the claim follows.
In the following theorem, we apply the previous lemma on classes of graphs which are closed under (spanning) subgraphs. In particular, general lower bounds for normal location-domination numbers in such classes also hold when we orient graphs.
Lemma 13. Let G be a class of graphs closed under subgraphs. If there exists a function f such that for each graph G ∈ G with n vertices we have

Proof. Assume by contradiction that
As proven in [27], planar graphs satisfy γ LD (G) ≥ n+10 7 and outerplanar graphs satisfy γ LD (G) ≥ 2n+3 7 . Since planar and outerplanar graphs are closed under subgraphs, the following is a consequence of Lemma 13. Let G ′ be an outerplanar graph on n vertices, then Lemma 15. Let G be a graph of order n. Then Proof. Let G be a graph of order n. In [31, Theorem 2] Slater has given a general lower bound for locating-dominating sets in regular graphs. Moreover, it is easy to generalize the proof for nonregular graphs giving . Thus, the claim follows from Lemma 13 with graph class G G = {H | H is a subgraph of G}.

Conjecture 2 holds for graph orientations
The main goal of this section is to prove Theorem 1 we restate here: Let G be a twin-free graph of order n with no isolated vertices, then We first need some auxiliary definitions and results. Let G = (V, E) be an undirected graph other than path P 2 . A vertex adjacent to a leaf is called a support vertex and a non-leaf, non-support vertex u which has only support vertices as neighbours is called a support link. The number of support vertices, leaves and support links in G are denoted by respectively s(G), ℓ(G) and sl(G). Moreover, let us denote by L(G), S(G) and SL(G) the sets of leaves, support vertices and support links, respectively, in G. By convention, for the path P 2 we assume that one of its two vertices is a support vertex and the other is a leaf.
2. There exists a minimum locating-dominating set C in T which contains all the vertices s ∈ S(T ) and for each s ∈ S(T ) there is exactly one leaf attached to s which is not in C.
Proof. Let C be a locating-dominating set. Let s ∈ S(T ). If s / ∈ C, then all the leaves attached to s are in C. Otherwise, C is not dominating. Let v be a leaf attached to s. We claim that C ′ = {s}∪C\{v} is a locating-dominating set. Indeed, we have I(C ′ ; v) = {s} and if I(C ′ ; u) = {s} for any v = u ∈ V \ C ′ , then I(C; u) = ∅. Thus, C ′ is a locating-dominating set. So, for every C, there exists a locating-dominating set of the same size containing s. We assume that S(T ) ⊆ C holds in the rest of the proof.
Assume by contradiction that Since C has minimum size, there exists a vertex u ∈ L(T ) ∪ C such that I(u) = {s} (otherwise v can be safely removed from C contradicting the minimality of C). However, if we now consider the set C ′ = {u} ∪ C \ {v}, then we notice immediately that C ′ is locating-dominating and the claim follows.
Locating-dominating sets in trees have been widely studied. Blidia et al. proved in [3] that Let us prove a slight improvement of this result that will be needed in the proof of the main result of this section.
Lemma 17. Let T be a tree of order n ≥ 2. Then Proof. Let T be a tree and let F = T − SL(T ). The set F induces a forest without isolated vertices. Moreover S(T ) = S(F ) and L(T ) = L(F ) (by choosing the right vertex in L and S if the component is a P 2 ). Let C be an optimal locating-dominating set in F such that S(F ) ⊆ C.
Observe that now C is also a locating-dominating set in T . Indeed, if u ∈ SL(T ), then I(u) ⊆ S(T ) and |I(u)| ≥ 2. Moreover, if I(v) = I(u), then we have a cycle. Finally, if u, v ∈ SL(T ), then The last inequality is due to Bound (1).
We are now ready to prove Theorem 1.
Proof of Theorem 1. Let T be a spanning tree of G such that ℓ(T ) − s(T ) is minimal among all the spanning trees of G. If T has ℓ(T ) = s(T ), then we are done by Lemma 12 and Lemma 17.
First, we claim that any leaf of T adjacent in T to a support vertex s such that |N (s)∩L(T )| ≥ 2, is adjacent, in G, only to vertices which are support vertices in T . Observe that if u and v are two leaves of T adjacent to the same support vertex s, then either u or v has another neighbour in G since G is twin-free. Moreover, if s ′ ∈ N G (u), then s ′ is a support vertex in T . Indeed, if s ′ is a leaf in T , then the spanning tree T ′ = T − us + us ′ satisfies ℓ(T ′ ) − s(T ′ ) < ℓ(T ) − s(T ), a contradiction with the minimality of T . Moreover, if s ′ is a non-leaf, non-support vertex, then we have s(T ′ ) = s(T ) + 1 and ℓ(T ′ ) = ℓ(T ), a contradiction.
We now construct an auxiliary graph G ′ as follows. First we add to the tree T every edge e = uv ∈ E(G) such that u ∈ L(T ), v ∈ S(T ) and there is a support vertex s ∈ S(T ) in N T (u) such that |N T (s) ∩ L(T )| ≥ 2. Then we delete some of the newly added extra edges so that there is exactly one leaf adjacent to every vertex in S(T ). The resulting graph is denoted by G ′ . Observe that, because G is twin-free, none of the vertices in L(T ) are pairwise twins in G ′ .
Let C ′ be an optimal locating-dominating set in T such that every support vertex is included in it and for each s ∈ S(T ) there exists a leaf u ∈ N (s) ∩ L(T ) such that u ∈ C ′ . By Lemma 16 such a set exists. Let us now denote C ′′ = C ′ \ L(T ). Now, Lemma 17 and Lemma 16 together imply that |C ′′ | ≤ n/2. Indeed, Let us denote I(u) = {u 1 , . . . , u t }, t ≥ 2, and assume without loss of generality that and w is not a leaf in T . Let us now consider the tree T ′′ = T − u 1 v + uu 2 . We notice that no new leaves are created since {w, u 2 } ⊆ N T ′′ (v) and u 1 has at least three neighbours in T , namely v, u and at least one other leaf. Moreover, the number of support vertices does not decrease. Indeed, u 2 ∈ S(T ′′ ) and u 1 ∈ S(T ′′ ). Finally, u ∈ L(T ) but u ∈ L(T ′′ ). Thus we have ℓ(T ′′ ) − s(T ′′ ) < ℓ(T ) − s(T ), a contradiction and hence, C is a locating-dominating set in G ′ , a spanning subgraph of G and the claim follows by Lemma 12.
This bound of n/2 is asymptotically tight even for graphs with large minimum degree as we will see in the next subsection (see Lemma 22). However, it can be improved in many cases, even without the twin-freeness assumption. Let us provide two simple classes for which we can improve it.
Remark 18. Let G be a graph on n vertices with a twin-free spanning subgraph G ′ with no isolated vertices. Then → γ LD (G ′ ) ≤ n/2 by Theorem 1 and by Lemma 12, we have γ LD (G) ≤ → γ LD (G ′ ). Hence, the existence of a twin-free spanning subgraph G ′ is enough for Theorem 1 to hold.
Lemma 19. Let G be a graph on n vertices with a Hamiltonian path. We have Proof. The Hamiltonian path is a spanning subgraph. Since γ LD (P n ) = 2n 5 as proven in [30], Lemma 12 ensures that → γ LD (G) ≤ 2n 5 . We say that a graph G has a P ≥t -factor (or t-path factor ) if it has a spanning subgraph containing only paths of length at least t as its components.

(Almost) regular graphs
The goal of this section is to prove that the n/2 bound can be drastically improved when the graph is (almost) regular. The proof is based on a probabilistic argument. Namely we prove that, if we select a random subset of vertices of the graph, then we can find an orientation where it is "almost" a locating-dominating set. That is, with positive probability, we can obtain a locating-dominating set from a random set by simply adding a small well-chosen subset of vertices to this random set.
A graph G is d-regular if all the vertices of G have degree exactly d. A class of graphs G is k-almost regular if for every graph G ∈ G, we have ∆(G) ≤ δ(G) k .
Theorem 21. Let G be a class of k-almost regular graphs. Then, there exists constant c G,k such that, for every G ∈ G, G satisfies Before proving Theorem 21, let us make a couple of remarks. First notice that the bound is tight up to a constant multiplicative factor since, by Theorem 7, Θ(log n) vertices are needed for cliques.
Another hypothesis of Theorem 21 asserts that there is a polynomial gap between the minimum and maximum degree. One can wonder if a similar result holds if we only have some assumptions on the minimum degree of the graph. We can prove that it is not true: Lemma 22. Let d, n 0 ∈ N, and ǫ > 0. There exists a twin-free graph G of minimum degree at least d, order n ≥ n 0 such that Proof. Let p and q be two integers with p ≥ q ≥ 4. We define the graph G p,q of order n = 4p + q as a disjoint union of p paths on four vertices complete to a set {v 1 , v 2 , ..., v q } of size q such that the subgraph induced by {v 1 , v 2 , ..., v q } is a cycle. An example is given by Figure 2. As p ≥ q the minimal degree is δ(G) = q + 1 and one can check G p,q is twin-free.
Let us prove that → γ LD (G p,q ) ≥ 2p − 2 4q+2 which is enough to obtain the lemma since then, → γ LD (G p,q )/n will tend to 1 2 , when p → ∞. Let D be an orientation of G p,q and let S be an optimal locating-dominating set of D. Let G 1 = G p,q [p 1 , p 2 , p 3 , p 4 ], G 2 = G p,q [q 1 , q 2 , q 3 , q 4 ] and G 3 = G p,q [r 1 , r 2 , r 3 , r 4 ] be three P 4 of G p,q which belongs to the disjoint union of P 4 's. If, for every 1 ≤ i ≤ 4 and every 1 ≤ j ≤ q, the edges p i v j , q i v j and r i v j have the same orientation in D, Indeed, if there is at most one vertex of S in each subgraph, then in each subgraph G i one extremity have no neighbour in G i ∩ S. Hence we can assume this is the case for p 1 and q 1 . Then p 1 and q 1 have the same neighbourhood in S, a contradiction. There are 2 q 4 orientations of edges between a set of four vertices and a set of q vertices so at least p − 2 × 2 q 4 = p − 2 q 4 +1 paths of the disjoint union contain at least two elements of S. So The rest of this section is devoted to prove Theorem 21. Let G be a graph in G. We can assume that G has minimum degree at least e 2 . (For graph of degree less than e 2 , the conclusion indeed follows since we can modify the constant to guarantee that c G,k · log δ δ · n is at least n). The proof is based on a probabilistic argument. We will select a subset of vertices at random and prove that, by only modifying it slightly (with high probability), we can construct an orientation of G such that this set is a locating-dominating set.
Let us first recall the Chernoff inequality.

Lemma 23.
[Chernoff ] Let X = n i=1 X i where X i = 1 with probability p and 0 otherwise and where all the X i are independent. Let µ = E(X) and r > 0. We have Also recall the Markov's inequality: If X is a random variable taking non-negative values and a > 0, then: In order to prove Theorem 21, we will also need the following general lemma: Lemma 24. Let G be a graph and X be a subset of vertices such that every vertex v not in X is adjacent to at least log ∆ + 1 vertices of X. Then there exists an orientation D of G where X is a locating-dominating set.
Proof. Let V ′ = {v 1 , . . . , v t } be an arbitrary ordering of V \ X. Let us prove that we can associate to each vertex v i of V ′ a non-empty subset S i of X ∩ N (v i ) such that, for every i = j, S i = S j . Let us prove that such a collection of sets S i can be found greedily. Since v 1 is adjacent to at least log ∆ + 1 ≥ 1 vertex of X, we can indeed find such a set for v 1 . Assume that we have already selected S 1 , . . . , S r . Let us prove that we can select a set for v r+1 . Let Y r+1 = N (v r+1 ) ∩ X and u ∈ Y r+1 . The number of subsets of Y r+1 containing u is 2 |Yr+1|−1 ≥ 2 log ∆+1 ≥ ∆. So at least one of them has not been selected since a subset S j can contain u only if v j u is an edge. We arbitrarily select a subset of Y r+1 containing u that is distinct from S 1 , . . . , S r , which completes the first part of the proof. Now, for every x ∈ X in N (v i ), we orient the edges from v i to x j if x / ∈ S i and orient from x to v i if x ∈ S i . One can easily check that X is a locating-dominating set of this orientation of the graph.
We now have all the ingredients to prove Theorem 21.
Proof of Theorem 21. Let us first start with the following claim: Claim 25. Let c ≥ 2 be constant. For every graph G of minimum degree δ, there exists a subset X of 25c · (log δ)/δ · n vertices 1 of G such that all the vertices of V \ X have at least c log δ neighbours in X.
Let us now enrich X with all the vertices u such that |X ∩ N (u)| is less than c log δ. By union bound, the average number of vertices that are added in X is at most n/δ 3 . Moreover, using again Markov inequality, we know that, with probability at least 1/2, the number of vertices that are added in X is at most 2 · n/δ 3 ≤ c · log δ δ · n. So, with probability at least 1/4, the size of X is at most 24c · log δ δ · n before X is enriched and we add at most c · log δ δ · n vertices in X during the second phase. So there exists a set X of size at most 25c log δ/δ · n such that all the vertices are either in X or have at least c log δ neighbours in X.
Let c = 2k. By Claim 25, G admits a subset of vertices X of size 50k · log δ/δ · n such that every vertex v is either in X or has at least 2k · log δ neighbours in X. We claim that we can orient the edges between X and V \ X to guarantee that all the vertices of V \ X have a different neighbourhood in X. It follows from Lemma 24 and the fact that log ∆ + 1 ≤ 2k · log δ since log ∆ ≤ k log δ.
Let us complete the results of this section with additional results on regular graphs or based on Lemma 24.
A set S is k-dominating in G if we have for each v ∈ V \ S that |N (v) ∩ S| ≥ k. Let us denote with γ k (G) the cardinality of a minimum k-dominating set of G. The following lemma is a simple consequence of Lemma 24.
Lemma 26. Let G be a graph with maximum degree ∆. Let k ≥ log ∆ + 1, then Since γ k (G) ≤ n − α(G) where k ≤ δ for graphs of minimum degree δ by [10,Corollary 14], the following is an immediate consequence of Lemma 26.
Corollary 27. Let G be a graph with maximum degree ∆ and minimum degree δ ≥ log ∆ + 1. Then A similar result holds for locating-dominating sets, when G is twin-free [16,Corollary 4.5].
Let M be a matching in a graph G. We say that a vertex u ∈ V (G) is M -unmatched if u is not an endpoint of any edge in M . there exists a set of adjacent edges which can be removed so that w has a unique I-set afterwards. Therefore, we may construct a spanning subgraph G ′′ with the property γ LD (G ′′ ) ≤ α ′ (G). Hence, the claim follows by Lemma 12.

Worst orientation
We now focus on the worst possible orientation. We again start with basic results. Then we study the lower bound → Γ LD (G) ≥ γ LD (G)/2 that we prove to be true for several classes of graphs and let it open in general. Finally, we consider lower bounds using the number of vertices.

Basic results
Let us start by first giving lower bounds that will be used all along the section. The maximum average degree of a graph G, denoted by mad(G) is the maximum quantity 2|E(H)| |V (H)| over all the subgraphs H of G.
Here the second inequality is due to Caro-Wei lower bound for independence number [5,32] and the last equality is due to equality ad(G) + ad(G) = n − 1.
All these bounds are tight: the first bound is tight for stars and the two others for complete graphs.
We still have → Γ LD (G) ≤ n − 1 as soon as G has at least one edge. As in the case of 2. G is a star; 3. G consists of a complete bipartite graph and possibly a single universal vertex.
Proof. By Theorem 3, we have → Γ LD (G) = n − 1 if n = 3 or G is a star. Moreover, since we consider oriented graphs, the third condition of Theorem 3 implies that C must be of size one. Thus, the claim follows.
Cycles on four vertices have a special role for best orientations. It is also the case for worst orientations, as illustrated by the following results.  Proof. Let G be a bipartite graph without C 4 . We have → Γ LD (G) ≥ α(G) by Lemma 29. By [25], we have α ′ (G) = β(G) since G is bipartite. Moreover, by [15], we have α(G) + β(G) = n.
Hence, α(G) = n − α ′ (G). Now, we have, by Lemma 33, Corollary 35. Let C n be a cycle on n vertices. Let n ≥ 3 and n = 4. We have Proof. By Lemma 32, we have by Corollary 34 applied to P n (where α(P n ) = n 2 ). Moreover, if we take a cyclic orientation of C n , the set of vertices with an odd index number forms an optimal locating-dominating set.
Observe that, for a path on n vertices, P n , we have γ LD (P n ) = ⌈2n/5⌉ since γ LD (P n ) = 2n 5 for paths [30] while we have α(P n ) = ⌈n/2⌉. As we mentioned above, there exist graphs without . However, we are not aware of any graph G without C 4 which does not attain the upper bound of Lemma 33.
Open problem 36. Does there exist a graph G without C 4 as a subgraph with

Lower bound with γ LD (G)
In Section 4.1, we have seen that → Γ LD (G) ≥ γ LD (G) if G is without C 4 subgraphs. One can easily remark that this equality does not hold in general. For example, for complete graphs we have γ LD (K n ) = n − 1 and → Γ LD (K n ) = ⌈n/2⌉ by Corollary 6. However the clique example is somehow unsatisfactory since all the vertices are twins. One can wonder if we can also provide an example of twin-free graphs where → Γ LD (G) < γ LD (G). We will prove (Theorem 42) that there exists graphs for which → Γ LD (G) < γ LD (G) is arbitrarily close from 1/2. Moreover, we strengthen the result that → Γ LD (G) ≥ γ LD (G) on graphs without C 4 to a wider class of graphs (Lemma 40). Despite our efforts, we were not able to find graphs for which → Γ LD (G) < γ LD (G)/2. We left as an open problem the following problem: Open problem 37. Let G be a graph, is it true that We were actually not able to prove the existence of any constant c such that, for any graph G, → Γ LD (G) ≥ c · γ LD (G). However, in the following theorem we present a bound with ∆(G).
Theorem 38. Let G be a graph. We have Proof. Let D be an orientation of G such that → Γ LD (G) = γ LD (D). Moreover, let S be an optimal locating-dominating set in D. Observe, that for each subset I of S, the set contains at most ∆(G) vertices: |S I | ≤ ∆(G). Let us next construct a new orientation D 1 by first taking for each set S I , ⌊|S I |/2⌋ disjoint vertex pairs within the set S I , that is, as many disjoint vertex pairs as possible. Then we number each vertex of V (G) as u i , 1 ≤ i ≤ |V (G)| so that each pair has consecutive numbers. Finally we orient each edge from u i to u j where i < j.
Let S 1 be an optimal locating-dominating set for orientation D 1 . Notice that |S 1 | ≤ → Γ LD (G). Moreover, S ′ 1 = S ∪ S 1 is a locating-dominating set in D and D 1 . Furthermore, S 1 separates each paired pair of vertices in D 1 . Thus, if for a pair u i , u i+1 , vertex x separates u i and u i+1 in D 1 , then either x = u i+1 or it separates also u i and u i+1 in G. Moreover, for each I ′ ⊆ S ′ 1 such that I = I ′ ∩ S, we have that S ′I ′ 1 ⊆ S I . Since S 1 separates the pairs in G, we have that If we now iterate this process ⌈log 2 (∆(G))⌉ times, each time creating a new orientation with a new numbering and a new optimal locating-dominating set for the orientation, then we finally get . Moreover, because we (almost) halve the number of vertices with the same I-set in G each time, no vertices in V \ S ′ t share the same I-set with the set S ′ t in G. Thus, S ′ t is locating-dominating in G and

Graphs for which
Lemma 39. Let G be a graph and D be an orientation of G such that no C 4 in G contains a directed path of length 4 in D. Then any locating-dominating set of D is a locating-dominating set of G. In particular, Proof. Let G be a graph and D be an orientation of G such that no C 4 in G contains a directed path of length 4 in D. Let S be locating-dominating in D. Let us assume that S is not locatingdominating in G. Set S is clearly dominating in G. Let v, u ∈ V \S be vertices with I G (v) = I G (u).
Since I D (v) = I D (u), we have |I G (v)| ≥ 2. Let us assume that c 1 ∈ I D (v) \ I D (u) and c 2 ∈ I D (u). But now we have a directed path c 2 uc 1 v, a contradiction.
Let M and R be subgraphs of G. An (M, R)-WORM colouring [17] of graph G, is a colouring of the vertices of G where no subgraph of G isomorphic to M is monochromatic and no subgraph of G isomorphic to R is heterochromatic (i.e. has all its vertices of different colours). The following lemma gives us a tool for applying Lemma 39.
Lemma 40. If G admits a (K 2 , C 4 )-WORM colouring, then Proof. Let G be a graph which admits a (K 2 , C 4 )-WORM colouring c using colours {1, . . . , k}. Let D be the orientation such that we have an edge from u to v if c(u) < c(v). Since c is a (K 2 , C 4 )-WORM colouring, it defines an orientation for any edge and We claim that no C 4 in D contains a directed path of length 4. Indeed, if there is a directed path u 1 u 2 u 3 u 4 , then c(u 1 ) < c(u 2 ) < c(u 3 ) < c(u 4 ) and if this path is contained in a C 4 , then this C 4 is heterochromatic, a contradiction. Then the claim follows from Lemma 39.
Observe that any proper colouring with at most three colours is also a (K 2 , C 4 )-WORM colouring. Hence, we get the following corollary (where χ(G) denotes the chromatic number of G).

Worst examples
The following theorem ensures that there exist examples of twin-free graphs where we almost reach the ratio 1 2 for → Γ LD (G)/γ LD (G). Theorem 42. There exists an infinite family of twin-free graphs G such that Proof. Let k, t ≥ 2 be integers and let H k,t be the graph with vertex set where we have 1 ≤ i ≤ k, 1 ≤ j ≤ t for each i and j. We illustrate graph H 3,3 in Figure 3. In other words, the set of vertices {v i,j | 1 ≤ i ≤ k} induces a clique V j t for every j. Similarly, the set of vertices {u i,j | 1 ≤ i ≤ k} induces a clique U j t for every j and the set of vertices {u i,j | 1 ≤ j ≤ t} induces a clique U i k for each i. In fact, the set of vertices u i,j , for 1 ≤ i ≤ k, 1 ≤ j ≤ t, forms the Cartesian product K t K k . Observe that H t,k is twin-free since each vertex u i,j has a unique neighbour v i,j and vice versa.
Let C be a locating-dominating set of H t,k . If we have {v i,j , u i,j , v i ′ ,j , u i ′ ,j } ∩ C = ∅. Then I(v i,j ) = I(v i ′ ,j ) and hence, we have a contradiction. Thus, γ LD (H t,k ) ≥ (k − 1)t. On the other hand, the set {v i,j | 1 ≤ i ≤ k, 1 ≤ j ≤ t} forms a locating-dominating set and hence, Let us then consider the oriented locating-dominating sets. Let D be an orientation of H t,k with → Γ LD (G) = γ LD (D). Let S j be a 2-dominating set in the tournament U j t for each j (i.e. each vertex outside S j is dominated twice). Let S ′ i be a dominating set in the tournament U i k \ t j=1 S j for each i in the orientation D. Observe that |S j | ≤ 2 log(t + 1) and |S ′ i | ≤ log(k + 1) for each i and j by [11]. Moreover, let C j be an optimal locating-dominating set in the tournament V j t . We have |C j | ≤ t/2.
Observe that for each i and j, vertex u i,j is now 3-dominated by k a=1 S ′ a ∪ t b=1 S b and I D (u i,j ) = I D (u i ′ ,j ′ ) where (i, j) = (i ′ , j ′ ). Moreover, each vertex v i,j is located by the set C j . Thus C is a locating-dominating set of D and → Γ LD (H k,t ) = γ LD (D) ≤ kt/2 + 2k log(t + 1) + t log(k + 1).
Finally, if we choose an orientation of H k,t such that each edge from v i,j is oriented to u i,j and such that all the cliques V j t are oriented transitively, we notice that we need at least t⌈k/2⌉ vertices in C.

Lower bound with the number of vertices
In this subsection, we consider how small → Γ LD (G) can be compared to the number of vertices. For the best orientation and the undirected case, there exist many graphs reaching the theoretical lower bound in Θ(log n) (see Theorem 7). For the worst orientation, we did not find any graph with → Γ LD (G) of order log n.
Open problem 43. Does there exist a class of graphs G such that for any G ∈ G on n vertices value → Γ LD (G) is logarithmic on n?
We have three reasons to believe there is a positive answer for Open problem 43. First, most of the other types of locating-dominating parameters can achieve logarithmic values on n. Secondly, we did not find a non-logarithmic lower bound. Thirdly, A natural class of candidates would be (Erdős-Renyi) random graphs where an unoriented locating-dominating set has indeed logarithmic size [14]. However, the worst orientation of such a graph is not easy to manipulate and then we were not able to study efficiently upper bounds on → Γ LD (G). On the other hand, in the following we give some properties which deny the possibility for a graph class G to have a logarithmic lower bound on n. Together with a well-known conjecture and an open problem, if they have a positive solution, these properties mean that if G has a certain type of a forbidden subgraph characterization, then it does not have a logarithmic lower bound for → Γ LD (G). In the following, we discuss these ideas and give some polynomial lower bounds for → Γ LD (G) in some graph classes.
Lemma 29 gives a linear lower bound for → Γ LD (G) in n for classes of graphs which have their chromatic number bounded by a constant since α(G) ≥ n/χ(G) and for classes of graphs with cliques of linear size. These results can be extended to obtain bounds in Ω(n β ) where β is a constant when a class of graphs G is χ-bounded by a polynomial function, that is, if there exists a polynomial function f such that χ(G) ≤ f (ω(G)) holds for all G ∈ G. Note that it has been asked [23] if it is true that every χ-bounded class admits a χ-bounding function that is polynomial. Moreover, Gyárfas [19] has conjectured that if the graph class G is F -free for some forest F , then G is χ-bounded.
Theorem 44 applies in particular for perfect graphs for which f is the identity function. Hence, if G is a perfect graph, then Theorem 44 can also be used to get a lower bound, for example, for claw-free graphs. In [8], the authors have shown that if G is a connected claw-free graph with an independent set of size at least 3, then χ(G) ≤ 2ω(G). Thus, → Γ LD (G) ≥ √ n/2. Similar idea works also for C 3 -free graphs. In [24], the author has shown that if G is C 3 -free, then α(G) ∈ Ω( √ n log n). Thus, also → Γ LD (G) ∈ Ω( √ n log n). Finally, we end the chapter by giving a class of perfect graphs which shows that Bound (2) is tight within a logarithmic multiplier. We denote by G H the cartesian product of G and H.

Proof. Let us denote the vertices of
There are 2m cliques, each of size m in G and every vertex belongs to exactly two of these cliques. We have ω(K m K m ) = χ(K m K m ) = m. Thus, m ≤ → Γ LD (K m K m ) and G is perfect. Let D be an orientation of G such that → Γ LD (G) = γ LD (D). Similarly, as in the proof of Theorem 42, we again construct a dominating set for each clique {(v i , u j ) | 1 ≤ i ≤ m} where j is fixed and a 2-dominating set for each clique {(v i , u j ) | 1 ≤ j ≤ m} where i is fixed. Observe that, in D, each dominating set has cardinality of at most log(m + 1) ( [11]) and hence, each 2-dominating set has cardinality of at most 2 log(m + 1). Since we have m dominating sets and m different 2-dominating sets, we have → Γ LD (K m K m ) ≤ 3m log(m + 1).