## The diameter of paired-domination vertex critical graphs.(English)Zbl 1174.05093

Summary: In this paper we continue the study of paired-domination in graphs introduced by Teresa W. Haynes and Peter J. Slater [Networks 32, 199-206 (1998; Zbl 0997.05074)]. A paired-dominating set of a graph  $$G$$ with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of $$G$$, denoted by $$\gamma _{\text{pr}}(G)$$, is the minimum cardinality of a paired-dominating set of  $$G$$. The graph  $$G$$ is paired-domination vertex critical if for every vertex  $$v$$ of  $$G$$ that is not adjacent to a vertex of degree one, $$\gamma _{\text{pr}}(G - v) < \gamma _{\text{pr}}(G)$$. We characterize the connected graphs with minimum degree one that are paired-domination vertex critical and we obtain sharp bounds on their maximum diameter. We provide an example which shows that the maximum diameter of a paired-domination vertex critical graph is at least $$\frac 32(\gamma _{\text{pr}}(G) - 2)$$. For $$\gamma _{\text{pr}}(G) \leq 8$$, we show that this lower bound is precisely the maximum diameter of a paired-domination vertex critical graph.

### MSC:

 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C35 Extremal problems in graph theory

Zbl 0997.05074
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### References:

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