On the size of identifying codes in triangle-free graphs

Abstract

In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a dominating set of $G$, and each vertex in $V(G)$ 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, Chakrabarty and Levitin in 1998. For a given identifiable graph $G$, let $\M(G)$ be the minimum cardinality of an identifying code in $G$. In this paper, we show that for any connected identifiable triangle-free graph $G$ on $n$ vertices having maximum degree $\Delta \ge 3$, $\M(G)\le n-\tfrac{n}{Δ+o(Δ)}$. This bound is asymptotically tight up to constants due to various classes of graphs including $(\Delta -1)$-ary trees, which are known to have their minimum identifying code of size $n-\frac{n}{\Delta -1+o(1)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound $\M(G)\le n-\tfrac{n}Δ+c$ holds for any nontrivial connected identifiable graph $G$.

References in corpus (1)

• Extremal graphs for the identifying code problem

Cited by in corpus (5)

• Identifying codes in line graphs
• Random subgraphs make identification affordable
• Strong Identification Codes for Graphs
• Identifying codes in graphs of given maximum degree: Characterizing trees
• Identifying codes of corona product graphs