Independent Domination Stable Trees and Unicyclic Graphs

Abstract

A set S ⊆ V ( G ) in a graph G is a dominating set if S dominates all vertices in G, where we say a vertex dominates each vertex in its closed neighbourhood. A set is independent if it is pairwise non-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 ID-stable if the independent domination number of G is not changed when any vertex is removed. In this paper, we study basic properties of ID-stable graphs and we characterize all ID-stable trees and unicyclic graphs. In addition, we establish bounds on the order of ID-stable trees.

Keywords

• independent domination
• stable graph
• tree
• unicyclic graph