Total pitchfork domination and its inverse in graphs

Abstract

New two domination types are introduced in this paper. Let [Formula: see text] be a finite, simple, and undirected graph without isolated vertex. A dominating subset [Formula: see text] is a total pitchfork dominating set if [Formula: see text] for every [Formula: see text] and [Formula: see text] has no isolated vertex. [Formula: see text] is an inverse total pitchfork dominating set if [Formula: see text] is a total pitchfork dominating set of [Formula: see text]. The cardinality of a minimum (inverse) total pitchfork dominating set is the (inverse) total pitchfork domination number ([Formula: see text]) [Formula: see text]. Some properties and bounds are studied associated with maximum degree, minimum degree, order, and size of the graph. These modified domination parameters are applied on some standard and complement graphs.