International audience

In the literature, several identification problems in graphs have been studied, of which, the most widely studied are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set. Such a (total-)dominating set endowed with a separation property is often referred to as a code of the graph. In this paper, we study the four separation properties location, closed-separation, open-separation and full-separation. We address the complexity of finding minimum separating sets in a graph and study the interplay of these separation properties with several codes (establishing a particularly close relation between separation and codes based on domination) as well as the interplay of separation and complementation (showing that location and full-separation are the same on a graph and its complement, whereas closed-separation in a graph corresponds to open-separation in its complement).

The problems of determining the minimum-sized identifying, locating-dominating and open locating-dominating codes of an input graph are special search problems that are challenging from both theoretical and computational viewpoints. In these problems, one selects a dominating set C of a graph G such that the vertices of a chosen subset of V(G) (i.e. either V(G) \ C or V(G) itself) are uniquely determined by their neighborhoods in C. A typical line of attack for these problems is to determine tight bounds for the minimum codes in various graph classes. In this work, we present tight lower and upper bounds for all three types of codes for block graphs (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or blocks) of a block graph and the order of the graph. Two of our upper bounds verify conjectures from the literature with one of them being now proven for block graphs in this article. As for the lower bounds, we prove them to be linear in terms of both the number of blocks and the order of the block graph. We provide examples of families of block graphs whose minimum codes attain these bounds, thus showing each bound to be tight.

An identifying code of a closed-twin-free graph G is a set S of vertices of G such that any two vertices in G have a distinct intersection between their closed neighborhoods and S. It was conjectured in [F. Foucaud, R. Klasing, A. Kosowski, A. Raspaud. On the size of identifying codes in triangle-free graphs. Discrete Applied Mathematics, 2012] that there exists an absolute constant c such that for every connected graph G of order n and maximum degree ∆, G admits an identifying code of size at most ∆-1/∆n + c. We provide significant support for this conjecture by proving it for the class of all bipartite graphs that do not contain any pairs of open-twins of degree at least 2. In particular, this class of bipartite graphs contains all trees and more generally, all bipartite graphs without 4-cycles. Moreover, our proof allows us to precisely determine the constant c for the considered class, and the list of graphs needing c ≥ 0. For ∆ = 2 (the graph is a path or a cycle), it is long known that c = 3/2 suffices. For connected graphs in the considered graph class, for each ∆ ≥ 3, we show that c = 1/∆ ≤ 1/3 suffices and that c is required to be positive only for a finite number of trees. In particular, for ∆ = 3, there are 12 trees with diameter at most 6 with a positive constant c and, for each ∆ ≥ 4, the only tree with positive constant c is the ∆-star. Our proof is based on induction and utilizes recent results from [F. Foucaud, T. Lehtilä. Revisiting and improving upper bounds for identifying codes. SIAM Journal on Discrete Mathematics, 2022].