Independent Domination Stable Trees and Unicyclic Graphs

Abstract

A set S ⊆ V ( G ) in a graph G is a dominating set if every vertex of G is either in S or adjacent to a vertex of S . A dominating set S is independent if any pair of vertices in S is not adjacent. The minimum cardinality of an independent dominating set on a graph G is called the independent domination number i ( G ) . A graph G is independent domination stable if the independent domination number of G remains unchanged under the removal of any vertex. In this paper, we study the basic properties of independent domination stable graphs, and we characterize all independent domination stable trees and unicyclic graphs. In addition, we establish bounds on the order of independent domination stable trees.