Bounds on the Identifying Codes in Trees
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In this paper, we continue the study of identifying codes in graphs, introduced by Karpovsky et al. (IEEE Trans Inf Theory 44:599–611, 1998). A subset S of vertices in a graph G is an identifying code if for every pair of vertices x and y of G, the sets \(N[x]\cap S\) and \(N[y]\cap S\) are non-empty and different. The minimum cardinality of an identifying code in G is denoted by M(G). We show that for a tree T with \(n\ge 3\) vertices, \(\ell \) leaves and s support vertices, \((2n-s+3)/4\le M(T) \le (3n+2\ell -1)/5\). Moreover, we characterize all trees achieving equality for these bounds.
KeywordsIdentifying code Tree
Mathematics Subject Classification05C69
The authors would like to thank both referees for their careful review and many helpful comments.