Domination Number of Vertex Amalgamation of Graphs

Abstract

For a graph G = (V, E), a subset S of V is called a dominating set if every vertex x in V is either in S or adjacent to a vertex in S. The domination number 𝜸(𝑮) is the minimum cardinality of the dominating set of G. The dominating set of G with a minimum cardinality denoted by 𝜸(𝑮)-set. Let G 1 , G 2 , ... , G t be subgraphs of the graph G. If the union of all these subgraphs is G and their intersection is {v}, then we say that G is the vertex-amalgamation of G 1 , G 2 , ... , G t at vertex v. Based on the membership of the common vertex v in the 𝜸(𝑮 there exist three conditions to be considered.