Independent Roman Domination Stable and Vertex-Critical Graphs

A
Roman 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 the
Roman domination number
of
$G$
. 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 the
independent Roman domination number
of
$G$
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 be
independent Roman domination vertex critical
or
$i_{R}$
-
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.