Paired-domination

Abstract

We are interested in dominating sets (of vertices) with the additional property that the vertices in the dominating set can be paired or matched via existing edges in the graph. This could model the situation of guards or police where each has a partner or backup. This paper will focus on those graphs in which the number of matched pairs of a minimum dominating set of this type equals the size of some maximal matching in the graph. In particular, we characterize the leafless graphs of girth seven or more of this type.

An introduction to proper-coupled-domination in graphs

“…If at least one of the vertex pairs {v 3 , v 4 } and {v 6t +1 , v 6t +2 } is not coupled, then without loss of generality, let us assume vertex pair {v 3 , v 4 } is not coupled. Now, consider P 6t+2 = C 6t+5 -{v 1 , v 2 , v 6t+5 } and let π′ be the coloring of the vertices of P 6t+2 with the same labels in C 6t+5 . We observe that neither of the end vertices v 3 and v 6t +4 of P 6t+2 is coupled with its support vertices in π′ of P 6t+2 , so by Lemma 7 we have γ cpl (P 6t+2 ; π′) < 4t + 2.…”

“…Let P 6t+2 = C 6t+5 -{v 1 , v 6t +4 , v 6t +5 } and let π′ be the coloring of the vertices of P 6t+2 with the same labels in C 6t+5 . We observe that neither of the end vertices v 1 and v 6t +4 is coupled with its support vertex in π′ of P 6t+2 , so by Lemma 7 we have γ cpl (P 6t+2 ;π′) < 4t+2.…”

“…The domination number γ(G) and the paired-domination number γ pr (G) are the minimum cardinalities of dominating sets and paired-dominating sets, respectively. Paired-dominating sets are considered in [1][2][3][4].…”

Domination Parameters in Coronene Torus Network

“…The paired-domination number γ pr (G) is the minimum cardinality of a paired-dominating set. If D is a paired-dominating set with a perfect matching M, then two vertices v j and v k are said to be paired in D if the edge v j v k ∈ M. Paired domination was introduced by Haynes and Slater [9] in 1998 and has been studied in [6,15]. Since every paired-dominating set is a total dominating set, γ (G) ≤ γ t (G) ≤ γ pr (G) for every graph with no isolated vertices [9].…”

The diameter of paired-domination vertex critical graphs

“…A PDS of cardinality γ pr (G) we call a γ pr (G)-set. Paired-domination was introduced by Haynes and Slater [14], [15] as a model for assigning backups to guards for security purposes, and is studied, for example, in [2], [3], [4], [6], [9], [13], [16], [17], [18], [19], [20] and elsewhere.…”