*Article* **Independent Domination Stable Trees and Unicyclic Graphs**

**Pu Wu 1, Huiqin Jiang 1, Sakineh Nazari-Moghaddam 2, Seyed Mahmoud Sheikholeslami 2, Zehui Shao 1,\* and Lutz Volkmann 3**


Received: 1 August 2019; Accepted: 2 September 2019; Published: 5 September 2019

**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 <sup>i</sup>*(*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
