**1. Introduction**

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. For every vertex *v* ∈ *<sup>V</sup>*(*G*), the *open neighborhood* of *v* is the set *NG*(*v*) = *<sup>N</sup>*(*v*) = {*u* ∈ *V*(*G*) | *uv* ∈ *<sup>E</sup>*(*G*)}, and its *closed neighborhood* is the set *NG*[*v*] = *<sup>N</sup>*[*v*] = *<sup>N</sup>*(*v*) ∪ {*v*}. The *degree* of a vertex *v* ∈ *V* is *dG*(*v*) = |*N*(*v*)|. A *leaf* of *G* is a vertex with degree one, and a *support vertex* is a vertex adjacent to a leaf. The set of all leaves adjacent to a vertex *v* is denoted by *<sup>L</sup>*(*v*). For two vertices *u* and *v*, the *distance dG*(*<sup>u</sup>*, *v*) from *u* to *v* is the number of the edges of a shortest *uv*-path in *G*. The *diameter* diam(*G*) of a graph *G* is the greatest distance among a pair of vertices of *G*. Assume *T* is a rooted tree and *v* ∈ *<sup>V</sup>*(*T*), let *<sup>C</sup>*(*v*) and *<sup>D</sup>*(*v*) denote the set of children and descendants of *v*, respectively, and *<sup>D</sup>*[*v*] = *<sup>D</sup>*(*v*) ∪ {*v*}. The *maximal subtree* at *v*, denoted by *Tv*, is the subgraph of *T* induced by *<sup>D</sup>*[*v*], and is denoted by *Tv*. For a graph *G*, let *I*(*G*) be the set of vertices with degree 1. The path and cycle on *n* vertices are denote by *Pn* and *Cn*, respectively.

A set *S* ⊆ *V* in a graph *G* is a *dominating set* if every vertex of *G* is either in *S* or adjacent to a vertex of *S*. The *domination number γ*(*G*) equals the minimum cardinality of a dominating set in *G*. There are many variants of the dominating set which are studied extensively, such as the independent dominating set [1], total domination [2,3], Roman domination [4,5], semitotal domination [6,7], etc. For a comprehensive treatment of domination in graphs, see the monographs by Haynes, Hedetniemi, and Slater [8,9].

A set is *independent* if it is pairwise non-adjacent. The minimum cardinality among all independent dominating sets on a graph *G* is called the *independent domination number i*(*G*) of *G*. An *<sup>i</sup>*(*G*)-set is an independent dominating set of *G* of cardinality *<sup>i</sup>*(*G*). This variation of graph domination has been studied extensively in the literature; see for example the books [8,9], and the readers can consult the new survey of Goddard and Henning [1].

The removal of a vertex from a graph can increase the independent domination number, decrease the independent domination number, or leave it unchanged. A graph *G* is independent domination vertex-critical or *i*-vertex-critical if *i*(*G* − *v*) < *i*(*G*) for every *v* ∈ *<sup>V</sup>*(*G*). The independent domination vertex-critical graphs have been studied by Ao [10] and Edwards [11] and elsewhere [12–14]. Here we focus on the case where the removal of any vertex leave the independent domination number unchanged.

A graph *G* is independent domination stable (ID-stable) if the independent domination number of *G* is not changed when any vertex is removed. 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 [2,15–17].

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.

We make use of the following results in this paper.

**Proposition 1** ([1])**.** *For n* ≥ 3*, i*(*Pn*) = *i*(*Cn*) = *n*3*.*

The next result is an immediate consequence of Proposition 1.

**Corollary 1.** *If n* ≥ 3*, then Cn is an ID-stable graph if and only if n* ≡ 1 (*mod* <sup>3</sup>)*.*

In the next sections, we will use the following notations: For a graph *G*, let:

> *W*(*G*) = {*u* ∈ *V*(*G*) | there exists an *<sup>i</sup>*(*G*)-set containing *u*}

and:

> *W*1,1(*G*) = {(*<sup>u</sup>*, *<sup>v</sup>*)|*<sup>u</sup>*, *v* ∈ *V*(*G*) and there exists an *<sup>i</sup>*(*G*)-set containing both of *u* and *<sup>v</sup>*}.
