Some new results on captive dominating sets in graphs

Abstract

A dominating set [Formula: see text] is said to be a captive dominating set of [Formula: see text] if [Formula: see text] has no isolated vertex ([Formula: see text] is a total dominating set) and each vertex [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. A captive dominating set [Formula: see text] is said to be a minimal captive dominating set if no proper subset [Formula: see text] of [Formula: see text] is a captive dominating set. The minimum cardinality of a minimal captive dominating set of [Formula: see text] is called the captive domination number of [Formula: see text] which is denoted by [Formula: see text]. In this paper, we have characterized some results, determine the values of the domination-related parameters for the graph and its splitting graph, relate captive domination and packing number of a graph, etc. We have also constructed graphs for which [Formula: see text]