New classification of graphs in view of the domination number of central graphs

Abstract

For a graph G, the central graph C(G) is the graph constructed from G by subdividing each edge of G with one vertex and also by adding an edge to every pair of non-adjacent vertices in G. Also for a graph G, let γ(G) and τ (G) be the domination number of G and the minimum cardinarity of a vertex cover of G, respectively. In this paper, we give a new classification of graphs concerning the domination number of central graphs and minimum vertex covers of graphs. Namely, we show that any graph G with at least three vertices can be classified into one of the two classes of graphs with γ(C(G)) = τ (G) and γ(C(G)) = τ (G) + 1, respectively, together with some special properties concerning a vertex cover of G. We also give some new results on the domination number of central graphs.