In an undirected graph , a subset such that is a dominating set of , and each vertex in is dominated by a distinct subset of vertices from , is called an identifying code of . The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph , let be the minimum cardinality of an identifying code in . In this paper, we show that for any connected identifiable triangle-free graph on vertices having maximum degree , . This bound is asymptotically tight up to constants due to various classes of graphs including -ary trees, which are known to have their minimum identifying code of size . We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant such that the bound holds for any nontrivial connected identifiable graph .
This research is supported by the ANR Project IDEA - Identifying coDes in Evolving grAphs, ANR-08-EMER-007, 2009–2011 and by the KBN Grant 4 T11C 047 25.