Domination Number of Vertex Amalgamation of Graphs

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 gamma(G) is the minimum cardinality of the dominating set of G. The dominating set of G with a minimum cardinality denoted by gamma(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 gamma(G(i))-set, there exist three conditions to be considered. First, if v elements of every gamma(G(i))-set, second if there is no gamma(G(i))-set containing v, and third if either v is element of gamma(G(i))-set for 1 <= i <= p or there is no gamma (G(i))-set containing v for p < i <= t. For these three conditions, the domination number of G as vertex-amalgamation of G(1), G(2),..., G(t) at vertex v can be determined.