Total Neighborhood Number in the Join and Corona of Graphs

Abstract

A set $S$ of vertices in a graph $G$ is called a \textit{neighborhood set} of $G$ if $G$ is the union of the subgraphs induced by the closed neighborhoods of the vertices in $S$. A subset $S \subseteq V(G)$ is a \textit{total neighborhood set} of $G$ if $S$ is a neighborhood set and every vertex $u \in V(G)$ is adjacent to at least one vertex $v \in S$. The \textit{neighborhood number} $n_0(G)$ (respectively, the \textit{total neighborhood number} $n_t(G)$) of $G$ is defined as the minimum cardinality of a neighborhood set (respectively, total neighborhood set) of $G$. In this paper, the author characterizes the class of graphs that attain the lower bound of the neighborhood number, which is two. Furthermore, the paper presents the characterization of neighborhood sets in the join of graphs and of total neighborhood sets in both the join and corona of graphs. Exact values for the neighborhood number of the join of graphs and for the total neighborhood number of the join and corona are also established.