Some bounds for the captive domination number of graphs

Abstract

A proper subset [Formula: see text] of vertices in the graph [Formula: see text] is a captive dominating set if it is a total dominating set and each vertex in [Formula: see text] dominates at least one vertex which does not belong to [Formula: see text]; also, the captive domination number of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum captive dominating set of [Formula: see text]. In [M. N. Al-Harere, A. A. Omran and A. T. Breesam, Captive domination in graphs, Discrete Math. Algorithms Appl. 12(6) (2020) 2050076], Al-Harere et al., presented some lower and upper bounds for the captive domination number of [Formula: see text], using the number of vertices and edges. Here, we present some upper bounds for [Formula: see text], using independence number and clique number. Also, the captive domination number of split graphs is studied. Among other results, some lower bounds for the captive domination number, in terms of maximum degree and the number of vertices with maximum degree, are presented.