Independent Roman Domination Number Research Articles

Independent Roman bondage of graphs

An independent Roman dominating function (IRD-function) on a graph G is a function f : V(G) → {0, 1, 2} satisfying the conditions that (i) every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2, and (ii) the set of all vertices assigned non-zero values under f is independent. The weight of an IRD-function is the sum of its function values over all vertices, and the independent Roman domination number iR(G) of G is the minimum weight of an IRD-function on G. In this paper, we initiate the study of the independent Roman bondage number biR(G) of a graph G having at least one component of order at least three, defined as the smallest size of set of edges F ⊆ E(G) for which iR(G − F) > iR(G). We begin by showing that the decision problem associated with the independent Roman bondage problem is NP-hard for bipartite graphs. Then various upper bounds on biR(G) are established as well as exact values on it for some special graphs. In particular, for trees T of order at least three, it is shown that biR(T) ≤ 3, while for connected planar graphs the upper bounds are in terms of the maximum degree with refinements depending on the girth of the graph.

Lower and upper bounds on independent double Roman domination in trees

For a graph G = ( V , E ) , a double Roman dominating function (DRDF) f : V → {0, 1, 2, 3} has the property that for every vertex v ∈ V with f ( v )=0 , either there exists a neighbor u ∈ N ( v ) , with f ( u )=3 , or at least two neighbors x , y ∈ N ( v ) having f ( x )= f ( y )=2 , and every vertex with value 1 under f has at least a neighbor with value 2 or 3 . The weight of a DRDF is the sum f ( V )=∑ v ∈ V f ( v ) . A DRDF f is an independent double Roman dominating function (IDRDF) if the vertices with weight at least two form an independent set. The independent double Roman domination number i d R ( G ) is the minimum weight of an IDRDF on G . In this paper, we show that for every tree T with diameter at least three, i ( T )+ i R ( T )−( s ( T ))/2 + 1 ≤ i d R ( T )≤ i ( T )+ i R ( T )+ s ( T )−2 , where i ( T ), i R ( T ) and s ( T ) are the independent domination number, the independent Roman domination number and the number of support vertex of T , respectively.

Further results on the independent Roman domination number of graphs

Let f : V (G) → {0, 1, 2} be a function on a graph G with vertex set V (G). Let Vi = {v ∈ V (G) : f (v) = i} for every i ∈ {0, 1, 2}. The function f is said to be an independent Roman dominating function on G if V 1 ∪ V 2 is an independent set and for every υ 2 V 0. The minimum weight among all independent Roman dominating functions f on G is the independent Roman domination number of G, and is denoted by iR (G). In this paper we continue with the study of this parameter. In particular, we provide new bounds on iR (G) in terms of other domination invariants. Some of our results are tight bounds that improve some well-known results. Finally, we compute the independent Roman domination number of some product graphs, and we provide an alternative proof to show that the problem of computing iR (G) is NP-hard.

Efficient algorithms for independent Roman domination on some classes of graphs

Let $G=(V,E)$ be a given graph of order $n $. A function $f : V to {0,1, 2}$ is an independent Roman dominating function (IRDF) on $G$ if for every vertex $vin V$ with $f(v)=0$ there is a vertex $u$ adjacent to $v$ with $f(u)=2$ and ${vin V:f(v)> 0}$ is an independent set. The weight of an IRDF $f$ on $G $ is the value $f(V)=sum_{vin V}f(v)$. The minimum weight of an IRDF among all IRDFs on $G$ is called the independent Roman domination number of~$G$. In this paper, we give algorithms for computing the independent Roman domination number of $G$ in $O(|V|)$ time when $G=(V,E)$ is a tree, unicyclic graph or proper interval graph.

Algorithmic aspects of outer independent Roman domination in graphs

Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman dominating function if every vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. If further the set [Formula: see text] is an independent set, then [Formula: see text] is called an outer independent Roman dominating function (OIRDF). Let [Formula: see text] and [Formula: see text]. Then [Formula: see text] is called the outer independent Roman domination number of [Formula: see text]. In this paper, we prove that the decision problem for [Formula: see text] is NP-complete for chordal graphs. We also show that [Formula: see text] is linear time solvable for threshold graphs and bounded tree width graphs. Moreover, we show that the domination and outer independent Roman domination problems are not equivalent in computational complexity aspects.

Algorithmic complexity of outer independent Roman domination and outer independent total Roman domination

Let $$G=(V,E)$$ be a graph. A function $$f : V \rightarrow \{0, 1, 2\}$$ is an outer independent Roman dominating function (OIRDF) on a graph G if for every vertex $$v \in V$$ with $$f (v) = 0$$ there is a vertex u adjacent to v with $$f (u) = 2$$ and $$\{x\in V:f(x)=0\}$$ is an independent set. The weight of f is the value $$ f(V)=\sum _{v\in V}f(v)$$ . An outer independent total Roman dominating function (OITRDF) f on G is an OIRDF on G such that for every $$v\in V$$ with $$f(v)>0$$ there is a vertex u adjacent to v with $$f (u)>0$$ . The minimum weight of an OIRDF on G is called the outer independent Roman domination number of G, denoted by $$\gamma _{oiR}(G)$$ . Similarly, the outer independent total Roman domination number of G is defined, denoted by $$\gamma _{oitR}(G)$$ . In this paper, we first show that computing $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) is a NP-hard problem, even when G is a chordal graph. Then, for a given proper interval graph $$G=(V,E)$$ we propose an algorithm to compute $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) in $${\mathcal {O}}(|V| )$$ time.

Independent Roman domination and 2-independence in trees

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. A Roman dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. A Roman dominating function [Formula: see text] is called an independent Roman dominating function if the set of all vertices with positive weights is an independent set. The weight of an independent Roman dominating function [Formula: see text] is the value [Formula: see text]. The independent Roman domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an independent Roman dominating function on [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a 2-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text]. The maximum cardinality of a 2-independent set of [Formula: see text] is the 2-independence number [Formula: see text]. These two parameters are incomparable in general, however, we show that for any tree [Formula: see text], [Formula: see text] and we characterize all trees attaining the equality.

Independent Roman Domination Number of Graphs

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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) . 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.

On the Unique Response Roman Domination Numbers of Graphs

Let G be a graph with vertex set V(G). A function f: V(G) → {0, 1, 2} with the ordered partition (V0, V1, V2) of V(G), where Vi = {V∈V(G) | f(V) = i} for i = 0, 1, 2, is a Roman dominating function if x ∈ V0 implies |N(x)∩V2|≥ 1. It is a unique response Roman function if x ∈ V0 implies |N(x) ≥ V2|≤ 1 and x ∈ V1 ∪ V2 implies that |N(x) ∩ V2| = 0. A function f: V(G) → {0, 1, 2} is a unique response Roman dominating function if it is both a unique response Roman function and a Roman dominating function. The unique response Roman domination number, denoted by uR(G), of G is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination of graphs, and provide some graphs whose unique response Roman domination number equals to the independent Roman domination number.

Trees with independent Roman domination number twice the independent domination number

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a RDF [Formula: see text] is the value [Formula: see text]. The Roman domination number, [Formula: see text], of [Formula: see text] is the minimum weight of a RDF on [Formula: see text]. An RDF [Formula: see text] is called an independent Roman dominating function (IRDF) if the set [Formula: see text] is an independent set. The independent Roman domination number, [Formula: see text], is the minimum weight of an IRDF on [Formula: see text]. In this paper, we study trees with independent Roman domination number twice their independent domination number, answering an open question.

Strong equality between the Roman domination and independent Roman domination numbers in trees

A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V −→ {0,1,2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF is the value f(V (G)) = P u2V (G) f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number R(G) (respectively, the independent Roman domination number iR(G)) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that R(G) strongly equals iR(G), denoted by R(G) ≡ iR(G), if every RDF on G of minimum weight is independent. In this paper we provide a constructive characterization of trees T with R(T) ≡ iR(T).

A note on the independent Roman domination in unicyclic graphs

A Roman dominating function (RDF) on a graph G = (V;E) is a function f : V ! f 0;1;2g satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF is the value f(V (G)) = P u2V (G) f(u): An RDF f in a graph G is independent if no two vertices as- signed positive values are adjacent. The Roman domination number R(G) (respectively, the independent Roman domination number iR(G)) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that R(G) strongly equals iR(G); denoted by R(G) iR(G); if every RDF on G of minimum weight is independent. In this note we characterize all unicyclic graphs G with R(G) iR(G): We consider finite, undirected, and simple graphs G with vertex set V = V (G) and edge set E = E(G). The open neighborhood of a vertex v 2 V is N(v) = NG(v) = fu 2 V j uv 2 Eg and the degree of v, denoted by dG(v), is the cardinality of its open neighborhood. A vertex of degree one is called a leaf, and its neighbor is called a support vertex. If v is a support vertex; then v is called strong if v is adjacent to at least two leaves. For a graph G, let f : V (G)! f0;1;2g be a function, and let (V0;V1;V2) be the ordered partition of V = V (G) induced by f, where Vi =fv2 V (G) : f(v) = ig for i = 0;1;2. There is a 1 1 correspondence between the functions f : V (G)!f0;1;2g and the ordered partitions (V0;V1;V2) of V (G). So we will write f = (V0;V1;V2). A function f : V (G)! f0;1;2g is a Roman dominating function (RDF) on G if every vertex u of G for which f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of an RDF is the value f(V (G)) = P u2V (G) f(u): An RDF f in a graph G is independent if no two vertices assigned positive values