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 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 G₁, G₂, ⋯, 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 G₁, G₂, ⋯, G_t at vertex v. Based on the membership of the common vertex v in the γ ( G_i )-set, there exist three conditions to be considered. First, if v elements of every γ ( G_i )-set, second if there is no γ ( G_i )-set containing v, and third if either v is element of γ ( G_i )-set for 1 ≤ i ≤ p or there is no γ ( G_i )-set containing v for p < i ≤ t . For these three conditions, the domination number of G as vertex-amalgamation of G₁, G₂, ⋯, G_t at vertex v can be determined.