Domination Number of Vertex Amalgamation of Graphs
Yayuk Wahyuni, Mohammad Imam Utoyo, Slamin Slamin · Journal of Physics Conference Series · 2017
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 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 G 1 , G 2 , ..., G t at vertex v can be determined.