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.
Highlights
Throughout this paper, V ( G ) and edge set E( G ) are used to denote the vertex set and edge set of G, respectively
Since T is an ID-stable tree, we deduce from Proposition 4 that for any vertex v ∈ V ( T 0 ), i ( T 0 − v ) + d T ( v3 ) − 1 = i ( T − v ) = i ( T ) = i ( T 0 ) + d T ( v3 ) − 1 and this implies that i ( T 0 − v) = i ( T 0 )
We show that G is an independent domination stable graph
Summary
Throughout this paper, V ( G ) and edge set E( G ) (briefly V, E) are used to denote the vertex set and edge set of G, respectively. The set of all leaves adjacent to a vertex v is denoted by L(v). T induced by D [v], and is denoted by Tv. For a graph G, let I ( G ) be the set of vertices with degree 1. The minimum cardinality among all independent dominating sets on a graph G is called the independent domination number i ( G ) of G. We focus on the case where the removal of any vertex leave the independent domination number unchanged. The domination stable problem consists of characterize graphs whose domination number (a type of domination number, e.g. total domination number, Roman domination number) remains unchanged under removal of any vertex or edge, or addition of any edge [2,15,16,17].
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