Excellent trees and secure domination

Abstract

A graph G is said to be γ-exccllent if each vertex of G is contained in some minimum dominating set of G. A dominating set S of G is a secure dominating set if for each v ε V(G) - S there exists a vertex u ε N(v)∩ S such that (S - {u}) ∪ {v} dominates G. The secure domination number γ,_s(G) of G is the smallest cardinality amongst all its secure dominating sets. We use a simple constructive characterisation of γ-excellent trees to obtain a constructive characterisation of trees with equal domination and secure domination numbers.