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.