An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G and every cycle contains at least three colors. We prove that every graph with maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4 colors, improving the previous bound of 9.62 (Delta - 1). Our bound results from the analysis of a very simple randomised procedure using the so-called entropy compression method. We show that the expected running time of the procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices and edges of G. Such a randomised procedure running in expected polynomial time was only known to exist in the case where at least 16 Delta colors were available. Our aim here is to make a pedagogic tutorial on how to use these ideas to analyse a broad range of graph coloring problems. As an application, also show that every graph with maximum degree Delta has a star coloring with 2 sqrt(2) Delta^{3/2} + Delta colors.Comment: 13 pages, revised versio

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to find a subset of vertices, normally called a solution set, using which all vertices of the graph are distinguished. The identification can be done by considering the neighborhood within the solution set, or by employing the distances to the solution vertices. Normally the goal is to minimize the size of the solution set then. Here we study the case of interval graphs, unit interval graphs, (bipartite) permutation graphs and cographs. For these classes of graphs we give tight lower bounds for the size of such solution sets depending on the order of the input graph. While such lower bounds for the general class of graphs are in logarithmic order, the improved bounds in these special classes are of the order of either quadratic root or linear in terms of number of vertices. Moreover, the results for cographs lead to linear-time algorithms to solve the considered problems on inputs that are cographs. * u, v if it (totally) dominates exactly one of them. A set S (totally) separates the vertices of a set X if all pairs of X are (totally) separated by a vertex of S. Whenever it is clear from the context, we will only say "separate" and omit the word "totally". We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [33,34]). A set S of vertices of a graph G is a locating-dominating set if it is a dominating set and it separates the vertices of V (G) \ S. The smallest size of a locating-dominating set of G is the location-domination number of G, denoted γ LD (G). Without the domination constraint, this concept has also been used under the name distinguishing set in [2] and sieve in [28].Definition 2 (Karpovsky, Chakrabarty and Levitin [27]). A set S of vertices of a graph G is an identifying code if it is a dominating set and it separates all vertices of V (G). The smallest size of an identifying code of G is the identifying code number of G, denoted γ ID (G). Definition 3 (Seo and Slater [31]). A set S of vertices of a graph G is an open locating-dominating set if it is a total dominating set and it totally separates all vertices of V (G). The smallest size of an open locating-dominating set of G is the open location-domination number of G, denoted γ OLD (G). This concept has also been called identifying open code in [25]. Separation could also be done using distances from the members of the solution set. Let d(x, u) denote the distance between two vertices x and u. Definition 4 (Harary and Melter [24], Slater [32]). A set B of vertices of a graph G is a resolving set if for each pair u, v of distinct vertices, there is a vertex x of B with d(x, u) = d(x, v). 1 The smallest size of a resolving set of G is the metric dimension of G, denoted dim(G).

An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from all other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand that if a graph on n vertices with at least one edge admits an identifying code, then a minimum identifying code has size at most n-1. Some classes of graphs whose smallest identifying code is of size n-1 were already known, and few conjectures were formulated to classify all these graphs. In this paper, disproving these conjectures, we classify all finite graphs for which all but one of the vertices are needed to form an identifying code. We also classify all infinite graphs needing the whole set of vertices in any identifying code. New upper bounds in terms of the number of vertices and the maximum degree of a graph are also provided

Abstract. An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We show a dichotomy for the size of the smallest identifying code in classes of graphs closed under induced subgraphs. Our dichotomy is derived from the VC-dimension of the considered class C, that is the maximum VC-dimension over the hypergraphs formed by the closed neighbourhoods of elements of C. We show that hereditary classes with infinite VC-dimension have infinitely many graphs with an identifying code of size logarithmic in the number of vertices while classes with finite VC-dimension have a polynomial lower bound.We then turn to approximation algorithms. We show that Min Id Code (the problem of finding a smallest identifying code in a given graph from some class C) is log-APX-hard for any hereditary class of infinite VCdimension. For hereditary classes of finite VC-dimension, the only known previous results show that we can approximate Min Id Code within a constant factor in some particular classes, e.g. line graphs, planar graphs and unit interval graphs. We prove that Min Id Code can be approximate within a factor 6 for interval graphs. In contrast, we show that Min Id Code on C 4 -free bipartite graphs (a class of finite VC-dimension) cannot be approximated to within a factor of c log(|V |) for some c > 0. [19]. For a complete survey on these results, the reader is referred to the online bibliography of Lobstein [24].Two vertices u and v are twins if. The whole vertex set V (G) is an identifying code if and only if G is twin-free. Since supersets of identifying codes are identifying, an identifying code exists for G if and only if it is twin-free. A natural problem in the study of identifying codes is to find one of a minimum size. Given a twin-free graph G, the smallest size of an identifying code of G is called the identifying code number of G and is denoted by γ ID (G). The problem of determining γ ID is called the Min Id Code problem, and its decision version is NP-complete [8]. Let X ⊆ V . We denote by G[X] the graph induced by the subset of vertices X. In this paper, we focus on hereditary classes of graphs, that is classes closed under taking induced subgraphs. We consider the two following problems: finding good lower bounds and approximation algorithms for the identifying code number.

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An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbors within the code. We study the edge‐identifying code problem, i.e. the identifying code problem in line graphs. If γID(G) denotes the size of a minimum identifying code of an identifiable graph G, we show that the usual bound γID(G)≥⌈log2(n+1)⌉, where n denotes the order of G, can be improved to Θ(n) in the class of line graphs. Moreover, this bound is tight. We also prove that the upper bound γID(scriptL(G))≤2|V(G)|−5, where L(G) is the line graph of G, holds (with two exceptions). This implies that a conjecture of R. Klasing, A. Kosowski, A. Raspaud, and the first author holds for a subclass of line graphs. Finally, we show that the edge‐identifying code problem is NP‐complete, even for the class of planar bipartite graphs of maximum degree 3 and arbitrarily large girth.

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W [2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable. * A short version of this paper, containing only the results about location-domination and metric dimension, appeared in the proceedings of the WG 2015 conference [28].pair in X has a vertex in S (totally) separating it. We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [52,53], Babai [3]). A set S of vertices of a graph G is a locating-dominating set if it is a dominating set and it separates the vertices of V (G) \ S.

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that appear in the closed neighborhood of u and v are distinct. Let χ lid (G) be the minimum number of colors used in a locally identifying vertex-coloring of G. In this paper, we give several bounds on χ lid for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether χ lid (G) = 3 for a subcubic bipartite graph G with large girth is an NP-complete problem.