Introduction of color class dominating sets in graphs

Abstract

Let \(G=(V, E)\) be a graph. In this paper, we define a new graph parameter called color class domination number of \(G\). A color class dominating set of \(G\) is a proper coloring \(\mathscr{C}\) of \(G\) with the extra property that every color class in \(\mathscr{C}\) is dominated by a vertex in \(G\). A color class dominating set is said to be a minimal color class dominating set if no proper subset of \(\mathscr{C}\) is a color class dominating set of \(G\). The color class domination number of \(G\) is the minimum cardinality taken over all minimal color class dominating sets of \(G\) and is denoted by \(\gamma_\chi(G)\). Here we also obtain \(\gamma_\chi(G)\) for Path graph, Cycle graph, Helm graph, Flower graph, Sunflower graph, Gear graph and Sunlet graph.