Just Excellence and Very Excellence in Graphs with Respect to Strong Domination

C. V. R. Harinarayanan ； C. Y. Ponnappan ； S. P. Subbiah ； R. Sundareswaran ； V. Swaminathan

《Tamkang Journal of Mathematics》 38卷2期 (2007/06) Pp. 167-175

並列摘要

A graph G is said to be excellent with respect to strong domination if each u∈V(G), belongs to some γ(subscript s)-set of G. G is said to be just excellent with respect to strong domination if each u∈V(G) is contained in a unique γ(subscript s)-set of G. A graph G which is excellent with respect to strong domination is said to be very excellent with respect to strong domination if there is a γ(subscript s)-set D of G such that to each vertex u∈V-D, there exists a vertex v∈D such that (D-{v})∪{u} is a γ(subscript s)-set of G. In this paper we study these two classes of graphs. A strong very excellent graph is said to be rigid very excellent with respect to strong domination if the following condition is satisfied. Let D be a very excellent γ(subscript s)-set of G. To each (The equation is abbreviated), let E(u, D)={v∈D: (D-{v})∪{u} is a γ(subscript s)-set of G}. If |E(u, D)|=1 for all (The equation is abbreviated) then D is said to be a rigid very excellent a γ(subscript s)-set of G. If G has at least one rigid very excellent γ(subscript s)-set of G then G is said to be a rigid very excellent graph with respect to strong domination (or) a strong rigid very excellent graph. Some results regarding strong very excellent graphs are obtained.