Some modified types of pitchfork domination and it's inverse

Abstract

Let G be a finite, simple graph, without isolated vertices. For any non-negative integers x and y, a set D ⊆ V is a ”pitchfork dominating set pds”, when every vertex in D, dominates at most y and at least x vertices of V − D. A subset D−1 of V − D is an inverse pds if it is a pitchfork set. The pitchfork domination number of G, γpf (G), is the number of elements of a smallest pds. The ”inverse pitchfork domination number” of G, γ −1 pf (G), is the number of elements of a smallest inverse pds. In this paper, some modified pitchfork dominations and its inverse dominations are introduced when x = 1 and y = 2. Several bounds and properties are given and proved. Then, these modified dominations are applied on some standard graphs such as: path, cycle, wheel, complete, complete bipartite graph and their complements.