Independent Roman Domination Stable and Vertex-Critical Graphs
Pu Wu, Zehui Shao, Enqiang Zhu, Huiqin Jiang, S. Nazari-Moghaddam, Seyed Mahmoud Sheikholeslami · IEEE Access · 2018
ARoman dominating function(RDF) on a graph$G$is a function$f: V(G) \rightarrow \{0, 1, 2\}$for which every vertex assigned 0 is adjacent to a vertex assigned 2. The weight of an RDF is the value$\omega (f) = \sum _{u \in V(G)}f(u)$. The minimum weight of an RDF on a graph$G$is called theRoman domination numberof$G$. An RDF$f$is called an independent Roman dominating function (IRDF) if the set$\{v\in V\mid f(v)\ge 1\}$is an independent set. The minimum weight of an IRDF on a graph$G$is called theindependent Roman domination numberof$G$and is denoted by$i_{R}(G)$. A graph$G$is independent Roman domination stable if the independent Roman domination number of$G$does not change under removal of any vertex. A graph$G$is said to beindependent Roman domination vertex criticalor$i_{R}$-vertex critical, if for any vertex$v$in$G$,$i_{R}(G-v) < i_{R}(G)$. In this paper, we characterize independent Roman domination stable trees and we establish upper bounds on the order of independent Roman stable graphs. Also, we investigate the properties of$i_{R}$- vertex critical graphs. In particular, we present some families of$i_{R}$-vertex critical graphs and we characterize$i_{R}$-vertex critical block graphs.