Analyzing Restrained Pitchfork Domination Across Path-Related Graph Structures

Abstract

Let G = (V,E) be a finite, simple, and undirected graph without an isolated vertex. A dominating subset D ⊆ V (G) is a restrained pitchfork dominating set if 1 ≤ |N(u) ∩ V − D| ≤ 2 for every u ∈ D and every vertex not in D is adjacent to at least one vertex in the same set. The cardinality of a minimum restrained pitchfork dominating set is the restrained pitchfork domination number γrpf (G). In the course of this investigation, we undertake an examination of the restrained pitchfork domination number within various path-related graphs. This analysis encompasses a range of graph structures, including the coconut tree, double star, banana tree, binomial tree, thorn path, thorn graph, and the square of the path denoted as Pn.