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 $V\left(G\right)$ 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 $\gamma \left(G\right)$ . Let $G_1$ and $G_2$ be disjoint graphs, $H_1$ be a subgraph of $G_1$ , $H_2$ be a subgraph of $G_2$ , and $f$ be an isomorphism from $H_1$ to $H_2$ . The amalgamation (the glued graph) of $G_1$ and $G_2$ at $H_1$ and $H_2$ with respect to $f$ is the graph $G=\underset{H_1{\cong }_f{H}_2}{G_1⊲⊳G_2}$ obtained by forming the disjoint union of $G_1$ and $G_2$ and then identifying $H_1$ and $H_2$ with respect to $f$ . In this paper, we determine the domination numbers of the amalgamations of two cycles at connected subgraphs.