Domination Numbers of Amalgamations of Cycles at Connected Subgraphs

Abstract

A set S of vertices of a graph G is a dominating set of G if every vertex in VG is adjacent to some vertex in S. A minimum dominating set in a graph G is a dominating set of minimum cardinality. The cardinality of a minimum dominating set is called the domination number of G and is denoted by γG. Let G1 and G2 be disjoint graphs, H1 be a subgraph of G1, H2 be a subgraph of G2, and f be an isomorphism from H1 to H2. The amalgamation (the glued graph) of G1 and G2 at H1 and H2 with respect to f is the graph G=G1⊲⊳G2H1≅fH2 obtained by forming the disjoint union of G1 and G2 and then identifying H1 and H2 with respect to f. In this paper, we determine the domination numbers of the amalgamations of two cycles at connected subgraphs.