Double Roman Domination Research Articles

Upper Bounds on the k-Tuple (Roman) Domination Number of a Graph

Rautenbach and Volkmann (Appl Math Lett 20:98–102, 2007), gave an upper bound for the k-tuple domination number of a graph. Rad (J Combin Math Comb Comput, 2019, in press) presented an improvement of the above bound using the Caro-Wei Theorem. In this paper, using the well-known Brooks’ Theorem for vertex coloring and vertex covers, we improve the above bounds on the k-tuple domination number under some certain conditions. In the special case $$k=1$$ , we improve the upper bounds for the domination number (Arnautov in Prikl Mat Program 11:3–8, 1974; Payan in Cahiers Centre Etudes Recherche Oper 17:307–317, 1975) and the Roman domination number (Cockayne et al. in Discrete Math 278:11–22, 2004). We also improve bounds given by Hansberg and Volkmann (Discrete Appl Math 157:1634–1639, 2009) for Roman k-domination number, and Rad and Rahbani (Discuss Math Graph Theory 39:41–53, 2019) for double Roman domination number.

Complexity Aspects of Variants of Independent Roman Domination in Graphs

For a simple, undirected graph $$G = (V,$$ E), an independent Roman dominating function (IRDF) $$f : V \rightarrow \lbrace 0, 1, 2 \rbrace $$ has the property that every vertex $$v \in V$$ with $$f(v) = 0$$ is adjacent to at least one vertex u, with $$f(u) = 2$$ , and no two vertices assigned positive values are adjacent. An independent Roman $$\lbrace 2 \rbrace $$ -dominating function (IR2DF) is an IRDF f with an additional property that, for every vertex $$v \in V$$ with $$f(v) = 0$$ , there exists at least two vertices $$x, y \in N_G(v)$$ with $$f(x) = f(y) = 1$$ . An independent double Roman dominating function (IDRDF) on G is a function $$f : V \rightarrow \lbrace 0, 1, 2, 3 \rbrace $$ , such that for every vertex $$v \in V$$ , if $$f(v) = 0$$ , then v has at least two neighbors x, y with $$f(x) = f(y) = 2$$ or one neighbor w with $$f(w) = 3$$ , and if $$f(v) = 1$$ , then v must have at least one neighbor w with $$f(w) \ge 2$$ , and no two vertices assigned positive values are adjacent. The weight of an IRDF (IR2DF, IDRDF) f is the sum $$f(V) = \sum _{v \in V}f(v)$$ . Given a graph G and a positive integer k, the independent Roman domination problem (IRDP), independent Roman $$\lbrace 2 \rbrace $$ -domination problem (IR2DP) and independent double Roman domination problem (IDRDP), respectively, is to check whether G has an IRDF, IR2DF, and IDRDF of weight at most k. The IRDP, IR2DP and IDRDP are known to be NP-complete for bipartite graphs. We investigate the complexity of these problems for dually chordal graphs and some subclasses of bipartite graphs, namely, star convex bipartite graphs and comb convex bipartite graphs. We also investigate the complexity of IR2DP and IDRDP for chordal graphs. The minimum independent Roman domination problem (MIRDP), minimum independent Roman $$\lbrace 2 \rbrace $$ -domination problem (MIR2DP), and minimum independent double Roman domination problem (MIDRDP), respectively, are to find an IRDF, IR2DF, and IDRDF of minimum weight in the input graph. We show that MIRDP, MIR2DP, and MIDRDP are linear time solvable for bounded tree-width graphs, chain graphs, and threshold graphs, a subclass of split graphs. We also show that the MIRDP, MIR2DP, and MIDRDP are APX-hard for graphs with bounded degree 4. Finally, we show that the domination problem and IRDP (IR2DP, IDRDP) are not equivalent in computational complexity aspects.

Some Properties of Double Roman Domination

A double Roman dominating function on a graph G is a function f:VG⟶0,1,2,3 satisfying the conditions that every vertex u for which fu=0 is adjacent to at least one vertex v for which fv=3 or two vertices v1 and v2 for which fv1=fv2=2 and every vertex u for which fu=1 is adjacent to at least one vertex v for which fv≥2. The weight of a double Roman dominating function f is the value fV=∑u∈Vfu. The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination numberγdRG of G. A graph with γdRG=3γG is called a double Roman graph. In this paper, we study properties of double Roman domination in graphs. Moreover, we find a class of double Roman graphs and give characterizations of trees with γdRT=γRT+k for k=1,2.

An Upper Bound on the Double Roman Domination Number

Let $$G=(V,E)$$ be a simple graph. A set $$M\subseteq E$$ is a matching if no two edges in M have a common vertex. The matching number, denoted $$\beta (G)$$ (or $$\beta $$ ), is the maximum size of a matching in G. A double Roman dominating function (DRDF) on a graph G is a function f: $$V\longrightarrow \{0,1,2,3\}$$ satisfying the conditions that for every vertex u of weight $$f(u)\in \left\{ 0,1\right\} $$ : $$\left( i\right) $$ if $$f(u)=0$$ , then u is adjacent to either at least one vertex v with $$f(v)=3$$ or two vertices $$v_{1}$$ , $$v_{2}$$ with $$f(v_{1})=f(v_{2})=2$$ . $$\left( ii\right) $$ if $$f(u)=1$$ , then u is adjacent to at least one vertex v with $$f(v)\in \left\{ 2,3\right\} $$ . The weight of a double Roman dominating function f is the value $$f(V)=\sum _{u\in V}f(u)$$ . The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number of G, denoted by $$\gamma _{dR}\left( G\right) $$ . In this paper, first, we note that $$\gamma _{dR}(G)\le 3\beta (G)$$ , where G is a graph without isolated vertices. Moreover, we give a descriptive characterization of block graphs G satisfying $$\gamma _{dR}(G)=3\beta (G)$$ . Finally, we show that the decision problem associated with $$\gamma _{dR}(G)=3\beta (G)$$ is $$CO-\mathcal {NP}$$ -complete for bipartite graphs.

The double Roman domination number of generalized Sierpiński graphs

In this paper, we study the double Roman domination number of generalized Sierpiński graphs [Formula: see text]. More precisely, we obtain a bound for the double Roman domination number of [Formula: see text]. We also find the exact value of [Formula: see text].

Independent double Roman domination in graphs

For a graph G = (V,E), a double Roman dominating function has the property that for every vertex with f(v) = 0, either there exists a vertex , with f(u) = 3, or at least two neighbors 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 . A DRDF f is called independent if the set of vertices with positive weight under f, is an independent set. The independent double Roman domination number is the minimum weight of an independent double Roman dominating function on G. In this paper, we show that for every graph G of order n, and , where and i(G) are the independent 3-rainbow domination, independent Roman domination and independent domination numbers, respectively. Moreover, we prove that for any tree G, .

On algorithmic complexity of double Roman domination

A double Roman dominating function (DRDF) on a graph G=(V,E) is a function f:V⟶{0,1,2,3} such that every vertex v∈V with f(v)=0 is either adjacent to a vertex u with f(u)=3 or two distinct vertices x and y with f(x)=f(y)=2, and every vertex v∈V with f(v)=1 is adjacent to a vertex u with f(u)≥2. The weight of f is the sum f(V)=∑v∈Vf(v). The minimum weight of a DRDF on G is the double Roman domination number of G, denoted by γdR(G). A graph G is a double Roman Graph if γdR(G)=3γ(G), where γ(G) is the domination number of G.In this paper, we first show that the decision problem associated to double Roman domination is NP-complete even when restricted to planar graphs. Then, we study the complexity issue of a problem posed in [R.A. Beeler, T.W. Haynesa and S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016), 23–29], and show that the problem of deciding whether a given graph is double Roman is NP-hard even when restricted to bipartite or chordal graphs. Then, we give linear algorithms that compute the domination number and the double Roman domination number of a given unicyclic graph. Finally, we give a linear algorithm that decides whether a given unicyclic graph is double Roman.

Complexity of Roman {2}-domination and the double Roman domination in graphs

For a simple, undirected graph a Roman {2}-dominating function (R2DF) has the property that for every vertex with f(v) = 0, either there exists a vertex with f(u) = 2, or at least two vertices with The weight of an R2DF is the sum The minimum weight of an R2DF is called the Roman {2}-domination number and is denoted by A double Roman dominating function (DRDF) on G is a function such that for every vertex if f(v) = 0, then v has at least two neighbors with or one neighbor w with f(w) = 3, and if f(v) = 1, then v must have at least one neighbor w with The weight of a DRDF is the value The minimum weight of a DRDF is called the double Roman domination number and is denoted by Given an graph G and a positive integer k, the R2DP (DRDP) problem is to check whether G has an R2DF (DRDF) of weight at most k. In this article, we first show that the R2DP problem is NP-complete for star convex bipartite graphs, comb convex bipartite graphs and bisplit graphs. We also show that the DRDP problem is NP-complete for star convex bipartite graphs and comb convex bipartite graphs. Next, we show that are obtained in linear time for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. Finally, we propose a -approximation algorithm for the minimum Roman {2}-domination problem and -approximation algorithm for the minimum double Roman domination problem, where Δ is the maximum degree of G.

Perfect double Roman domination of trees

For a graph G with vertex set V(G) and function f:V(G)→{0,1,2,3}, let Vi be the set of vertices assigned i by f. A perfect double Roman dominating function of a graph G is a function f:V(G)→{0,1,2,3} satisfying the conditions that (i) if u∈V0, then u is either adjacent to exactly two vertices in V2 and no vertex in V3 or adjacent to exactly one vertex in V3 and no vertex in V2; and (ii) if u∈V1, then u is adjacent to exactly one vertex in V2 and no vertex in V3. The perfect double Roman domination number of G, denoted γdRp(G), is the minimum weight of a perfect double Roman dominating function of G. We prove that if T is a tree of order n≥3, then γdRp(T)≤9n∕7. In addition, we give a family of trees T of order n for which γdRp(T) approaches this upper bound as n goes to infinity.

Critical concept for double Roman domination in graphs

A double Roman dominating function (DRDF) on a graph [Formula: see text] is a function [Formula: see text] such that (i) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least two vertices assigned a [Formula: see text] or to at least one vertex assigned a [Formula: see text] and (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] The weight of a DRDF is the sum of its function values over all vertices. The double Roman domination number [Formula: see text] equals the minimum weight of a DRDF on [Formula: see text] The concept of criticality with respect to various operations on graphs has been studied for several domination parameters. In this paper, we study the concept of criticality for double Roman domination in graphs. In addition, we characterize double Roman domination edge super critical graphs and we will give several characterizations for double Roman domination vertex (edge) critical graphs.

Roman [formula omitted]-domination (double Italian domination)

For a graph G=(V,E) with V=V(G) and E=E(G), a Roman {3}-dominating function is a function f:V→{0,1,2,3} having the property that ∑u∈NG(v)f(u)≥3, if f(v)=0, and ∑u∈NG(v)f(u)≥2, if f(v)=1 for any vertex v∈G. The weight of a Roman {3}-dominating function f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a Roman {3}-dominating function on G is the Roman {3}-domination number of G, denoted by γ{R3}(G). We initiate the study of Roman {3}-domination and show its relationship to domination, Roman domination, Roman {2}-domination (Italian domination) and double Roman domination. Finally, we present an upper bound on the Roman {3}-domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we show that associated decision problem for Roman {3}-domination is NP-complete, even for bipartite graphs.

A linear algorithm for double Roman domination of proper interval graphs

A function [Formula: see text] is a double Roman dominating function on a graph [Formula: see text] if for every vertex [Formula: see text] with [Formula: see text] either there is a vertex [Formula: see text] with [Formula: see text] or there are distinct vertices [Formula: see text] with [Formula: see text] and for every vertex [Formula: see text] with [Formula: see text] there is a vertex [Formula: see text] with [Formula: see text]. The weight of a double Roman dominating function [Formula: see text] on [Formula: see text] is the value [Formula: see text]. The minimum weight of a double Roman dominating function on [Formula: see text] is called the double Roman domination number of [Formula: see text]. In this paper, we give an algorithm to compute the double Roman domination number of a given proper interval graph [Formula: see text] in [Formula: see text] time.

Integer linear programming formulations for double roman domination problem

ABSTRACT For a graph , a double Roman dominating function (DRDF) is a function having the property that if , then vertex v must have at least two neighbours assigned 2 under f or at least one neighbour u with , and if , then vertex v must have at least one neighbour u with . In this paper, we consider the double Roman domination problem, which is an optimization problem of finding the DRDF f such that is minimum. We propose five integer linear programming (ILP) formulations and one mixed integer linear programming formulation with polynomial number of constraints for this problem. Some additional valid inequalities and bounds are also proposed for some of these formulations. Further, we prove that the first four models indeed solve the double Roman domination problem, and the last two models are equivalent to the others regardless of the variable relaxation or usage of a smaller number of constraints and variables. Additionally, we use one ILP formulation to give an -approximation algorithm. All proposed formulations and approximation algorithm are evaluated on randomly generated graphs to compare the performance.

Algorithmic results on double Roman domination in graphs

Given a graph $$G=(V,E)$$, a function $$f:V\longrightarrow \{0,1,2,3\}$$ is called a double Roman dominating function on G if (i) for every $$v\in V$$ with $$f(v)=0$$, there are at least two neighbors of v that are assigned 2 under f or at least a neighbor of v that is assigned 3 under f, and (ii) for every vertex v with $$f(v)=1$$, there is at least one neighbor w of v with $$f(w)\ge 2$$. The weight of a double Roman dominating function f is $$f(V)=\sum _{u\in V}f(u)$$. The double Roman domination number of G, denoted by $$\gamma _{dR}(G)$$ is the minimum weight of a double Roman dominating function on G. For a graph $$G=(V,E)$$, Min-Double-RDF is to find a double Roman dominating function f with $$f(V)=\gamma _{dR}(G)$$. The decision version of Min-Double-RDF is shown to be NP-complete for chordal graphs and bipartite graphs. In this paper, we first strengthen the known NP-completeness of the decision version of Min-Double-RDF by showing that the decision version of Min-Double-RDF remains NP-complete for undirected path graphs, chordal bipartite graphs, and circle graphs. We then present linear time algorithms for computing the double Roman domination number in proper interval graphs and block graphs. We then discuss on the approximability of Min-Double-RDF and present a 2-approximation algorithm in 3-regular bipartite graphs.

On the Global Double Roman Domination of Graphs

We establish some inequalities between the global double Roman domination number $$\gamma _\mathrm{gdR}(G)$$ and double Roman domination number $$\gamma _\mathrm{dR}(G)$$ of graphs G. We also completely characterize the trees T with $$\gamma _\mathrm{gdR}(T)=\gamma _\mathrm{dR}(T)+k$$ for $$k\in \{0,1,2,3\}$$, which partially answer an open problem posed by Shao et al. (J Discrete Math Sci Cryptogr 22:31–44, 2019).

On the Independent Double Roman Domination in Graphs

An independent double Roman dominating function (IDRDF) on a graph $$G=(V,E)$$ is a function $$f{:}V(G)\rightarrow \{0,1,2,3\}$$ having the property that if $$f(v)=0$$ , then the vertex v has at least two neighbors assigned 2 under f or one neighbor w assigned 3 under f, and if $$f(v)=1$$ , then there exists $$w\in N(v)$$ with $$f(w)\ge 2$$ , such that the set of vertices with positive weight is independent. The weight of an IDRDF is the value $$\sum _{u\in V}f(u)$$ . The independent double Roman domination number $$i_\mathrm{dR}(G)$$ of a graph G is the minimum weight of an IDRDF on G. We continue the study of the independent double Roman domination and show its relationships to both independent domination number (IDN) and independent Roman $$\{2\}$$ -domination number (IR2DN). We present several sharp bounds on the IDRDN of a graph G in terms of the order of G, maximum degree and the minimum size of edge cover. Finally, we show that, any ordered pair (a, b) is realizable as the IDN and IDRDN of some non-trivial tree if and only if $$2a + 1 \le b \le 3a$$ .

English

for a graph G = (V, E), a double Roman dominating function is a function f: V → {0, 1, 2, 3} having the property that if f (v) = 0, then the vertex v must have at least two neighbors assigned 2 under f or one neighbor with f (w) = 3, and if f (v) = 1, then the vertex v must have at least one neighbor with f (w) ⩾ 2. The weight of a double Roman dominating function f is the sum $$f(V) = \sum\nolimits_{v \in V} {f(v)} $$. The minimum weight of a double Roman dominating function on G is called the double Roman domination number of G and is denoted by γdR(G). In this paper, we establish a new upper bound on the double Roman domination number of graphs. We prove that every connected graph G with minimum degree at least two and G ≠ C5 satisfies the inequality $${\gamma _{{\rm{dR}}}}(G)\leqslant \left\lfloor {{\textstyle{{13} \over {11}}}n} \right\rfloor $$. One open question posed by R. A. Beeler et al. has been settled.

Independent Double Roman Domination in Graphs

For a graph $$G = (V, E)$$, a double Roman dominating function (DRDF) on G is a function $$f : V \rightarrow \{0, 1, 2, 3\}$$ having the property that if $$f(v) = 0$$, then vertex v has at least two neighbors assigned 2 under f or one neighbor w with $$f(w)=3$$, and if $$f(v)=1$$, then vertex v has at least one neighbor w with $$f(w)\ge 2$$. A DRDF f is called an independent double Roman dominating function (IDRDF) if the set of vertices with positive weight is independent. The weight of an IDRDF is the sum $$f (V) =\sum _{v\in V} f (v)$$. The independent double Roman domination number $$i_{dR}(G)$$ is the minimum weight of an IDRDF on G. In this paper, we initiate the study of independent double Roman domination. We first show that the decision problem associated with $$i_{dR}(G)$$ is NP-complete for bipartite graphs and then we present some sharp bounds on the independent double Roman domination number.

Outer independent double Roman domination

An outer independent double Roman dominating function (OIDRDF) of a graph G is a function h from V(G) to {0, 1, 2, 3} for which each vertex with label 0 is adjacent to a vertex with label 3 or at least two vertices with label 2, and each vertex with label 1, is adjacent to a vertex with label greater than 1; and all vertices labeled by 0 is independent. The weight of an OIDRDF h is ∑w ∈ V(G)h(w), and the outer independent double Roman domination number γoidR(G) is the minimum weight of an OIDRDF on G. In this article, we provide various bounds on γoidR(G) and we show that its determining is NP-complete on chordal and bipartite graphs. Moreover, we establish Nordhaus–Gaddum bounds for γoidR(G)+γoidR(G¯).

An improved upper bound on the double Roman domination number of graphs with minimum degree at least two

A double Roman dominating function (DRDF) on a graph G=(V,E) is a function f:V→{0,1,2,3} having the property that if f(v)=0, then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f(w)=3, and if f(v)=1, then vertex v must have at least one neighbor w with f(w)≥2. The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number γdR(G) is the minimum weight of a DRDF on G. Let G be a connected graph G of order n and minimum degree at least two. With the exception of seven graphs of order at most seven, Beeler et al. (2016) observed that γdR(G)≤6n5 and posed the question whether this bound can be improved. Amjadi et al. (2018) settled this question by proving that γdR(G)≤8n7. In this paper, we improve this bound to γdR(G)≤11n10. Moreover, we provide an infinite family of graphs attaining this bound.