Publications Repository - Gdańsk University of Technology

In an undirected graph G, a subset C⊆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 gammaID(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 Δ≥3, gammaID(G)<=n - n/(Delta+o(Delta)). This bound is asymptotically tight up to constants due to various classes of graphs including (Δ−1)-ary trees, which are known to have their minimum identifying code of size n - 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 gammaID(G)<=n - n/Delta + c holds for any nontrivial connected identifiable graph G.

Authors

• Florent Foucaud,
• Ralf Klasing,
• dr inż. Adrian Kosowski link open in new tab ,
• Andre Raspaud