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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

Citations:

Zbl 0997.05074
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