Injective Eccentric Domination in Graphs

Abstract

The concept of domination has inspired researchers which has contributed to a vast literature on domination. A subset of is said to be a dominating set, if every vertex not in is adjacent to at least one vertex in . The eccentricity of is the distance to a vertex farthest from . Thus . For a vertex each vertex at a distance from is an eccentric vertex. The eccentric set of a vertex is defined as . Let , then is known as an eccentric point set of if for every , has at least one vertex such that . A dominating set is called an eccentric dominating set if it is also an eccentric point set. In this article the concept of injective eccentric domination is introduced for simple, connected and undirected graphs. An eccentric dominating set is called an injective eccentric dominating set if for every vertex there exists a vertex such that where is the set of vertices different from and , that are adjacent to both and . Theorems to determine the exact injective eccentric domination number for the basic class of graphs are stated and proved. Nordhaus-Gaddum results are proposed. The injective eccentric dominating set, injective eccentric domination number , upper injective eccentric dominating set and upper injective eccentric domination number for different standard graphs are tabulated.