Eccentric Domination and Restrained Eccentric Domination in Circulant Graphs Cpá2, 3ñ

Abstract

A subset D of the vertex set V(G) of a graph G is said to be a dominating set if every vertex not in D is adjacent to at least one vertex in D. A dominating set D is said to be an eccentric dominating set if for every vÎV-D, there exists at least one eccentric vertex of v in D. The minimum cardinality of an eccentric dominating set is called the eccentric domination number and is denoted by g_ed(G). A subset D of V(G) is a restrained eccentric dominating set if D is a restrained dominating set of G and for every v Î V - D, there exists at least one eccentric vertex of v in D. The minimum of the cardinalities of the restrained eccentric dominating set of G is called the restrained eccentric domination number of G and is denoted by g_red(G). Let p ³ 6 be a positive integer. The circulant graph C_pá2, 3ñ is the graph with vertex set  {v₀, v₁, v₂, …, v_p-1} and edge set {{v_i, v_i+j}: iÎ{0, 1, 2, …, p-1} and jÎ{2, 3}}.  In this paper, we initiate the study of domination number, restrained domination number, eccentric domination number and restrained eccentric domination number in the circulant graphs C_pá2, 3ñ.