On the total closed neighbourhood graph of a graph
V. R. Kulli, Nanda S. Warad · Journal of Discrete Mathematical Sciences and Cryptography · 2001
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 ui and 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) ⋃ V 1(G) with two points as adjacent if they correspond to two adjacent points of G or one corresponds to a point ui of V 1(G) and the other to a point wj of G where wj is 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.