On the total closed neighbourhood graph of a graph

Abstract

Let V(G) be the set of points of G. The open neighbourhood N(u) of a point u. in V(G) is the set of points adjacent to u. The closed neighbourhood N[u] of a point u in V(G) is given by For each point vi of G, we take a new point uiand the resulting set of points is denoted by V 1(G). The total closed neighbourhood graph Ntc(G) of a graph G is defined as the graph having point set V(G) ? V1(G) with two points as adjacent if they correspond to two adjacent points of G or one corresponds to a point uiof V1(G) and the other to a point wjof G where wjis in N[vi]. In this paper, we present characterization of graphs whose total closed neighbourhood graphs are planar, outerplanar, minimally nonouterplanar. We give characterizations of graphs with planar total closed neighbourhood graphs and outerplanar total closed neighbourhood graphs in terms of forbidden subgraphs. © 2001 Taylor & Francis Group, LLC.