Upper eccentric domination in graphs

Abstract

A set S ⊆ V is said to be a dominating set in a graph G if every vertex in V – S is adjacent to some vertex in S and the domination number of G is defined to be the minimum cardinality of a dominating set in G. A dominating set D is said to be an eccentric dominating set if for every v ∈ V–D, there exists at least one eccentric vertex of v in D. An eccentric dominating set D is a minimal eccentric dominating set if no proper subset S ⊂ D is an eccentric dominating set. In this paper, we introduce a new parameter known as upper eccentric domination number. The upper eccentric domination number ⎾ed(G) of a graph G equals the maximum cardinality of a minimal eccentric dominating set, that is ⎾ed(G) = max |D|, where the maximum is taken over D in D’, where D’ is the set of all minimal eccentric dominating sets of G.