The Monopoly in the Join of Graphs (Journal of Informatics and Mathematical Sciences, 2018)
In a graph \(G = (V,E)\), a set \(M\subseteq V(G)\) is said to be a monopoly set of \(G\) if every vertex \(v\in V-M\) has, at least, \(\frac{d(v)}{2}\) neighbors in \(M\). The monopoly size \(mo(G)\) of \(G\) is the minimum cardinality of a monopoly set among all monopoly sets of \(G\). A join graph is the complete union of two arbitrary graphs. In this paper, we investigate the monopoly set in the join of graphs. As consequences the monopoly size of the join of graphs is obtained. Upper and lower bound of the monopoly size of join graphs are obtained. The exact values of monopoly size for the join of some standard graphs with others are obtained.
Keywords: Monopoly set; Monopoly size; Join of graphs; Monopoly size of join of graphs