On the size of identifying codes in triangle-free graphs
Résumé
In an undirected graph $G=(V,E)$, a subset $C\subseteq V$ 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 $\M(G)$ 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 $\Delta\ge 2$, $\M(G)\le n-\frac{n}{3(\Delta+1)}$.
Origine : Fichiers produits par l'(les) auteur(s)