Changing and unchanging 2-rainbow independent domination

Abstract

For a function $f:V(G)\to \{0,1,2\}$ we denote by $V_i$ the set of vertices to which the value $i$ is assigned by $f$, i.e. $V_i=\{x\in V(G):f(x)=i\}$. If a function $f:V(G)\to \{0,1,2\}$ satisfying the condition that $V_i$ is independent for $i\in \{1,2\}$ and every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=i$ for each $i\in \{1,2\}$, then $f$ is called a 2-rainbow independent dominating function (2RiDF). The weight $w(f)$ of a 2RiDF $f$ is the value $w(f)=|V_1|+|V_2|$. The minimum weight of a 2RiDF on a graph $G$ is called the \emph{2-rainbow independent domination number} of $G$. A graph $G$ is 2-rainbow independent domination stable if the 2-rainbow independent domination number of $G$ remains unchanged under removal of any vertex. In this paper, we characterize 2-rainbow independent domination stable trees and we study the effect of edge removal on 2-rainbow independent domination number in trees.