Graphs and Combinatorics

, Volume 35, Issue 3, pp 599–609

# Bounds on the Identifying Codes in Trees

• Seyed Masoud MirRezaei
Original Paper

## Abstract

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.

## Keywords

Identifying code Tree

05C69

## Notes

### Acknowledgements

The authors would like to thank both referees for their careful review and many helpful comments.

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