On the size of identifying codes in triangle-free graphs

Florent Foucaud; Ralf Klasing; Adrian Kosowski; Andre Raspaud

doi:10.1016/j.dam.2012.02.009

On the size of identifying codes in triangle-free graphs

Abstract

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.

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DOI:: Digital Object Identifier (open in new tab) 10.1016/j.dam.2012.02.009
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Category:: Articles
Type:: artykuł w czasopiśmie wyróżnionym w JCR
Published in:: DISCRETE APPLIED MATHEMATICS no. 160,
ISSN: 0166-218X
Language:: English
Publication year:: 2012
Bibliographic description:: Foucaud F., Klasing R., Kosowski A., Raspaud A.: On the size of identifying codes in triangle-free graphs// DISCRETE APPLIED MATHEMATICS. -Vol. 160, nr. iss. 10-11 (2012),
DOI:: Digital Object Identifier (open in new tab) 10.1016/j.dam.2012.02.009
Verified by:: Gdańsk University of Technology