On the size of identifying codes in triangle-free graphs

Details On the size of identifying codes in triangle-free graphs On the size of identifying codes in triangle-free graphs

2010-10-28

arxiv.org/abs/1010.5975v2

Computer Science

Discrete Mathematics

Scientific paper

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 $\Delta\geq 3$, $\M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}$. 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 $n-\tfrac{n}{\Delta-1+o(1)}$. We conjecture that the bound $\M(G)\le n-\tfrac{n}{\Delta}+O(1)$ holds for any nontrivial connected identifiable graph $G$.

Affiliated with

Also associated with

No associations

Rating

On the size of identifying codes in triangle-free graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the size of identifying codes in triangle-free graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the size of identifying codes in triangle-free graphs will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-584010