On the size of identifying codes in triangle-free graphs

In an undirected graph G = (V,E), a subset C ⊆ V such that C is a dominating set of G, and each vertex in V is dominated by a distinct subset of vertices from C, is called an identifying code of G. The concept of identifying codes was introduced by Karpovsky et al. in 1998. Because of the variety of its applications, for example for fault-detection in networks or the location of fires in facilities, it has since been widely studied. For a given graph G, let γ(G) be the minimum cardinality of an identifying code in G. In this paper, we show that for any connected triangle-free graph G on n vertices having maximum degree ∆ ≥ 2, γ(G) ≤ n− n 3(∆+1) .