Identifying Codes in Hereditary Classes of Graphs and VC-Dimension

Abstract

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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