Journal of Combinatorial Optimization

, Volume 35, Issue 2, pp 613–631

# On computing a minimum secure dominating set in block graphs

• Anupriya Jha
Article

## Abstract

In a graph $$G=(V,E)$$, a set $$D \subseteq V$$ is said to be a dominating set of G if for every vertex $$u\in V{\setminus }D$$, there exists a vertex $$v\in D$$ such that $$uv\in E$$. A secure dominating set of the graph G is a dominating set D of G such that for every $$u\in V{\setminus }D$$, there exists a vertex $$v\in D$$ such that $$uv\in E$$ and $$(D{\setminus }\{v\})\cup \{u\}$$ is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite graphs.

## Keywords

Domination Secure domination Block graphs Linear time algorithm NP-complete

## Notes

### Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments leading to improvements in the presentation of the paper.

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