Double Roman Dominating Function Research Articles

New bounds on the outer-independent total double Roman domination number

A double Roman dominating function (DRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying (i) if [Formula: see text] then there must be at least two neighbors assigned two under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text]; and (ii) if [Formula: see text] then [Formula: see text] must be adjacent to a vertex [Formula: see text] such that [Formula: see text]. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on [Formula: see text] if the set of vertices labeled [Formula: see text] induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its function values over all vertices, and the outer-independent total Roman domination number [Formula: see text] is the minimum weight of an OITDRDF on [Formula: see text]. In this paper, we establish various bounds on [Formula: see text]. In particular, we present Nordhaus–Gaddum-type inequalities for this parameter. Some of our results improve the previous results.

DOUBLE ROMAN DOMINATION NUMBER OF MIDDLE GRAPH

For any graph G(V, E), a function f : V (G) 0, 1, 2, 3 is called Double Roman dominating function (DRDF) if the following properties holds, If f (v) = 0, then there exist two vertices v1, v2 ∈ N (v) for which f (v1) = f (v2) = 2 or there exist one vertex u ∈ N (v) for which f (u) = 3.∈ If f (v) = 1, then there exist one vertex u N (v) for which f (u) = 2 or Σ f (u) = 3. The weight of DRDF is the value w(f ) = v∈V (G) f (v). The minimum weight among all double Roman dominating function is called double Roman domination number and is denoted by γdR(G). In this article we initiated research on double Roman domination number for middle graphs. We established lower and upper bounds and also we characterize the double Roman domination number of middle graphs. Later we calculated numerical value of double Roman domination number of middle graph of path, cycle, star, double star and friendship graphs.

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.

Inverse double Roman domination in graphs

For a graph [Formula: see text], a double Roman dominating function (DRDF) is a function [Formula: see text] such that each vertex [Formula: see text] with [Formula: see text] is adjacent to at least two vertices labeled [Formula: see text] or one vertex labeled [Formula: see text] and each 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 [Formula: see text] is the sum of all labelings [Formula: see text] and is denoted by [Formula: see text]. If [Formula: see text] is a DRDF on [Formula: see text] with minimum weight [Formula: see text], then its inverse double Roman dominating function (IDRDF) [Formula: see text] is a DRDF on [Formula: see text], such that [Formula: see text], where [Formula: see text]. The inverse double Roman domination number (IDRDN) of [Formula: see text], denoted by [Formula: see text] is the minimum weight of such a function. We introduce this new type of inverse dominating function, obtain some bounds for the IDRDN of [Formula: see text]. We characterize the graphs having [Formula: see text] and the highest. We also present an approach for constructing graphs with the desired IDRDN.

Further results on independent double roman trees

A double Roman dominating function (DRDF) on a graph is a function such that every vertex u with f(u) = 0 is adjacent to at least one vertex assigned a 3 or to at least two vertices assigned a 2, and every vertex v with f(v) = 1 is adjacent to at least one vertex assigned 2 or 3. The weight of a DRDF is the sum of its function values over all vertices. A DRDF f is an independent double Roman dominating function (IDRDF) if the set of vertices assigned 1, 2 and 3 is independent. The independent double Roman domination number is the minimum weight over all IDRDFs of G. Every graph G satisfies where i(G) is the independent domination number. In this paper, we give a characterization of all trees T with

Algorithm and hardness results in double Roman domination of graphs

Let G=(V,E) be a graph. A double Roman dominating function (DRDF) of G is a function f:V→{0,1,2,3} such that (i) each vertex v with f(v)=0 is adjacent to either a vertex u with f(u)=3 or two vertices u1 and u2 with f(u1)=f(u2)=2, and (ii) each vertex v with f(v)=1 is adjacent to a vertex u with f(u)>1. The double Roman domination number of G is the minimum weight of a DRDF along all DRDFs on G, where the weight of a DRDF f on G is f(V)=∑v∈Vf(v). In this paper, we first propose an algorithm to compute the double Roman domination number of an interval graph G=(V,E) in O(|V|+|E|) time, answering a problem posed in Banerjee et al. (2020) [2]. Next, we show that the decision problem associated with the double Roman domination is NP-complete for split graphs. Finally, we show that the computational complexities of the Roman domination problem and the double Roman domination problem are different.

On the double Roman bondage numbers of graphs

For a graph [Formula: see text], a double Roman dominating function (DRDF) is a function [Formula: see text] having the property that if [Formula: see text] for some vertex [Formula: see text], then [Formula: see text] has at least two neighbors assigned [Formula: see text] under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text], and if [Formula: see text] then [Formula: see text] has at least one neighbor [Formula: see text] with [Formula: see text]. The weight of a DRDF [Formula: see text] is the sum [Formula: see text]. The minimum weight of a DRDF on a graph [Formula: see text] is the double Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. The double Roman bondage number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all edge subsets [Formula: see text] such that [Formula: see text]. In this paper, we study the double Roman bondage number in graphs. We determine the double Roman bondage number in several families of graphs, and present several bounds for the double Roman bondage number. We also study the complexity issue of the double Roman bondage number and prove that the decision problem for the double Roman bondage number is NP-hard even when restricted to bipartite graphs.

Total double Roman domination numbers in digraphs

Let [Formula: see text] be a finite and simple digraph with vertex set [Formula: see text]. A double Roman dominating function (DRDF) on digraph [Formula: see text] is a function [Formula: see text] such that every vertex with label 0 has an in-neighbor with label 3 or two in-neighbors with label 2 and every vertex with label 1 have at least one in-neighbor with label at least 2. The weight of a DRDF [Formula: see text] is the value [Formula: see text]. A DRDF [Formula: see text] on [Formula: see text] with no isolated vertex is called a total double Roman dominating function if the subgraph of [Formula: see text] induced by the set [Formula: see text] has no isolated vertex. In this paper, we initiate the study of the total double Roman domination number in digraphs and show its relationship to other domination parameters. In particular, we present some bounds for the total double Roman domination number and we determine the total double Roman domination number of some classes of digraphs.

Double Roman Domination in Generalized Petersen Graphs

The double Roman domination can be described as a strengthened defense strategy. In an empire, each city can be protected by at most three troops. Every city having no troops must be adjacent to at least two cities with two troops or one city with three troops. Every city having one troop must be adjacent to at least one city with more than one troop. Such an assignment is called a double Roman dominating function (DRDF) of an empire/a graph. The minimum number of troops under such an assignment is the double Roman domination number, denoted as $$\gamma _{dR}$$ . Shao et al. (2018) determine the exact value of $$\gamma _{dR}(P(n,1))$$ . Jiang et al. (2018) determine $$\gamma _{dR}(P(n,2))$$ . In this article, we investigate the double Roman domination number of P(n, k) for $$k\ge 3$$ . We determine the exact value of $$\gamma _{dR}(P(n,k))$$ for $$n\equiv 0(\bmod 4)$$ and $$k\equiv 1(\bmod 2)$$ , and present an improved upper bound of $$\gamma _{dR}(P(n,k))$$ for $$n\not \equiv 0(\bmod 4)$$ or $$k\not \equiv 1(\bmod 2)$$ . Our results imply P(n, 3) for $$n\equiv 0(\bmod 4)$$ is double Roman which can partially answer the open question present by Beeler et al. (2016).

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.

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.

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.

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.

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.

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.

Global double Roman domination in graphs

A double Roman dominating function (DRDF) on a graph G = (V, E) is a function f : V(G) → {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. A DRDF f is called a global double Roman dominating function (GDRDF) if f is also a DRDF of the complement of G. The weight of a GDRDF is the sum of its function value over all vertices. The global double Roman domination number of G, denoted by γgdR(G), is the minimum weight of a GDRDF on G. In this paper, we initiate the study of the global double Roman domination number. We obtain some properties of global double Roman domination number.

On the Total Double Roman Domination

Let G = (V, E) be a simple graph. A double Roman dominating function (DRDF) on G is a function f from the vertex set V of G into {0, 1, 2, 3} such that if f (u) = 0, then u must have at least two neighbors assigned 2 or one neighbor assigned 3 under f , and if f (u) = 1, then u must have at least one neighbor assigned at least 2 under f . The weight of a DRDFf is the value f (V) = Σu∈V(G)f (u). The total double Roman dominating function (TDRDF) on a graph G without isolated vertices is a DRDFf on G with the additional condition that the subgraph of G induced by the set {v ∈ V : f (v) ≥ 1} is isolated-free. The total double Roman domination number ytdR(G) is the smallest weight among all TDRDFs on G. In this paper, we first show that the decision problem for the total double Roman domination is NP-hard for chordal and bipartite graphs, and then we establish some sharp bounds on total double Roman domination number.

The Double Roman Domination Numbers of Generalized Petersen Graphs P(n, 2)

A double Roman dominating function (DRDF) f on a given graph G is a mapping from V ( G ) to { 0 , 1 , 2 , 3 } in such a way that a vertex u for which f ( u ) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f ( u ) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w ( f ) = ∑ u ∈ V ( G ) f ( u ) . The minimum weight of a DRDF on a graph G is called the double Roman domination number γ d R ( G ) of G. In this paper, we determine the exact value of the double Roman domination number of the generalized Petersen graphs P ( n , 2 ) by using a discharging approach.

An upper bound on the double Roman domination number

A double Roman dominating function (DRDF) on a graph $$G=(V,E)$$ is a function $$f : V \rightarrow \{0, 1, 2, 3\}$$ 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 $$f(w)\ge 2$$ . The weight of a DRDF f is the value $$f(V) = \sum _{u \in V}f(u)$$ . The double Roman domination number $$\gamma _{dR}(G)$$ of a graph G is the minimum weight of a DRDF on G. Beeler et al. (Discrete Appl Math 211:23–29, 2016) observed that every connected graph G having minimum degree at least two satisfies the inequality $$\gamma _{dR}(G)\le \frac{6n}{5}$$ and posed the question whether this bound can be improved. In this paper, we settle the question and prove that for any connected graph G of order n with minimum degree at least two, $$\gamma _{dR}(G)\le \frac{8n}{7}$$ .