Construction of Dominating Sets of Certain Graphs

Abstract

Let G be a simple graph. A set S ⊆ V is a dominating set of G , if every vertex in V ∖ S is adjacent to at least one vertex in S . We denote the family of dominating sets of a graph G with cardinality i by 𝒟 ( G , i ) . In this paper we introduce graphs with specific constructions, which are denoted by G ( m ) . We construct the dominating sets of G ( m ) by dominating sets of graphs G ( m − 1 ) , G ( m − 2 ) , and G ( m − 3 ) . As an example of G ( m ) , we consider 𝒟 ( P n , i ) . As a consequence, we obtain the recursive formula for the number of dominating sets of G ( m ) .