Double Roman Dominating Research Articles

Outer Independent Double Roman Domination Stability in Graphs

An outer independent double Roman dominating function (OIDRDF) on a graph G is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that (i) if f ( v ) = 0 , then the 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 the vertex v must have at least one neighbor w with f ( w ) ≥ 2 and (ii) the subgraph induced by the vertices assigned 0 under f is edgeless. The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number γ o i d R ( G ) is the minimum weight of an OIDRDF on G . The γ o i d R -stability ( γ − o i d R -stability, γ + o i d R -stability) of G , denoted by s t γ o i d R ( G ) ( s t − γ o i d R ( G ) , s t + γ o i d R ( G ) ), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the outer independent double Roman domination number. In this paper, we determine the exact values on the γ o i d R -stability of some special classes of graphs, and present some bounds on s t γ o i d R ( G ) . In addition, for a tree T with maximum degree Δ , we show that s t γ o i d R ( T ) = 1 and s t − γ o i d R ( T ) ≤ Δ , and characterize the trees that achieve the upper bound.

Signed total double Roman dominating functions in graphs

A signed total double Roman dominating function (STDRDF) on an isolated-free graph G = ( V , E ) is a function f : V ( G ) → { − 1 , 1 , 2 , 3 } such that (i) every vertex v with f ( v ) = − 1 has at least two neighbors assigned 2 under f or one neighbor w with f(w) = 3, (ii) every vertex v with f(v) = 1 has at least one neighbor w with f ( w ) ≥ 2 and (iii) ∑ u ∈ N ( v ) f ( u ) ≥ 1 holds for any vertex v. The weight of a STDRDF is the value f ( V ( G ) ) = ∑ u ∈ V ( G ) f ( u ) . The signed total double Roman domination number γ sdR t ( G ) is the minimum weight of a STDRDF on G. In this article, we provide various bounds on γ sdR t ( G ) and we show that the corresponding decision problem is NP-complete for bipartite and chordal graphs. In addition, we determine the signed total double Roman domination number of some classes of graphs including cycles, complete graphs and complete bipartite graphs.

Quasi total double Roman domination in graphs

A quasi total double Roman dominating function (QTDRD-function) on a graph G = ( V ( G ) , E ( G ) ) is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that (i) 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; (ii) if f(v) = 1, then vertex v has at least one neighbor w with f ( w ) ≥ 2 , and (iii) if x is an isolated vertex in the subgraph induced by the set of vertices assigned nonzero values, then f(x) = 2. The weight of a QTDRD-function f is the sum of its function values over the whole vertices, and the quasi total double Roman domination number γ qtdR ( G ) equals the minimum weight of a QTDRD-function on G. In this paper, we first show that the problem of computing the quasi total double Roman domination number of a graph is NP-hard, and then we characterize graphs G with small or large γ qtdR ( G ) . Moreover, we establish an upper bound on the quasi total double Roman domination number and we characterize the connected graphs attaining this bound.

Double Roman Domination in Cartesian Product

Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then there exist $ v_{1},v_{2}\in N(v)$ such that $f(v_{1})=2=f(v_{2})$ or there exists $ w \in N(v)$ such that $f(w)=3$, and if $f(v)=1$, then there exists $ w \in N(v)$ such that $f(w)\geq 2$ is called a double Roman dominating function (DRDF). The weight of a DRDF $f$ is the sum $f(V)=\sum_{v\in V}f(v)$, and the minimum among the weights of DRDFs on $G$ is the double Roman domination number, $\gamma_{dR}(G)$, of $G$. In this paper, we study the impact of cartesian product on the double Roman domination number.

Majority double Roman domination in graphs

A majority double Roman dominating function (MDRDF) 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 with 2 or to at least one vertex [Formula: see text] with [Formula: see text], (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] and (iii) [Formula: see text], for at least half of the vertices in [Formula: see text]. The weight of an MDRDF is the sum of its function values over all vertices. The majority double Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text]. In this paper, we introduce and study the majority double Roman domination number on some classes of graphs.

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.

On Double Roman Dominating Functions in Graphs

Let G be a connected graph. A function f : V (G) → {0, 1, 2, 3} is a double Roman dominating function of G if for each v ∈ V (G) with f(v) = 0, v has two adjacent vertices u and w for which f(u) = f(w) = 2 or v has an adjacent vertex u for which f(u) = 3, and for each v ∈ V (G) with f(v) = 1, v is adjacent to a vertex u for which either f(u) = 2 or f(u) = 3. The minimum weight ωG(f) = P v∈V (G) f(v) of a double Roman dominating function f of G is the double Roman domination number of G. In this paper, we continue the study of double Roman domination introduced and studied by R.A. Beeler et al. in [2]. First, we characterize some double Roman domination numbers with small values in terms of the domination numbers and 2-domination numbers. Then we determine the double Roman domination numbers of the join, corona, complementary prism and lexicographic product of graphs.

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.

Restrained condition on double Roman dominating functions

We continue the study of restrained double Roman domination in graphs. For a graph G=(V(G),E(G)), a double Roman dominating function f is called a restrained double Roman dominating function (RDRD function) if the subgraph induced by {v∈V(G)∣f(v)=0} has no isolated vertices. The restrained double Roman domination number (RDRD number) γrdR(G) is the minimum weight ∑v∈V(G)f(v) taken over all RDRD functions of G.We first prove that the problem of computing γrdR is NP-hard even for planar graphs, but it is solvable in linear time when restricted to bounded clique-width graphs such as trees, cographs and distance-hereditary graphs. Relationships between γrdR and some well-known parameters such as restrained domination number γr, domination number γ or restrained Roman domination number γrR are investigated in this paper by bounding γrdR from below and above involving γr, γ or γrR for general graphs, respectively. We prove that γrdR(T)≥n+2 for any tree T≠K1,n−1 of order n≥2 and characterize the family of all trees attaining the lower bound. The characterization of graphs with small RDRD numbers is given in this paper.

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

Twin signed double Roman domination numbers in directed graphs

Let [Formula: see text] be a finite simple directed graph (shortly digraph). A function [Formula: see text] is called a twin signed double Roman dominating function (TSDRDF) if (i) every vertex [Formula: see text] with [Formula: see text] has at least two in-neighbor assigned a 2 or at least an in-neighbor [Formula: see text] with [Formula: see text], also at least two out-neighbor assigned a 2 or at least an out-neighbor [Formula: see text] with [Formula: see text] (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least an in-neighbor and an out-neighbor [Formula: see text] with [Formula: see text] and (iii) [Formula: see text] and [Formula: see text] hold for any vertex [Formula: see text]. The weight of a TSDRDF [Formula: see text] is [Formula: see text], the twin signed double Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of a TSDRDF on [Formula: see text]. In this paper, we initiate the study of twin signed double Roman domination in digraphs and we present some bounds for [Formula: see text].

Restrained double Roman domination of a graph

For a graph G = (V, E), a restrained 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 w with f(w) = 3, and if f(v) = 1, then the vertex v must have at least one neighbor w with f(w) ≥ 2, and at the same time, the subgraph G[V0] which includes vertices with zero labels has no isolated vertex. The weight of a restrained double Roman dominating function f is the sum f(V) = ∑v∈V f(v), and the minimum weight of a restrained double Roman dominating function on G is the restrained double Roman domination number of G. We initiate the study of restrained double Roman domination with proving that the problem of computing this parameter is NP-hard. Then we present an upper bound on the restrained double Roman domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. We study the restrained double Roman domination versus the restrained Roman domination. Finally, we investigate the bounds for the restrained double Roman domination of trees and determine trees T attaining the exhibited bounds.

An improved upper bound on the independent double Roman domination number of trees

For a graph an independent double Roman dominating function (IDRDF) is a function having the property that: (i) every vertex with f(v) = 0 has a neighbor u with f(u) = 3 or at least two neighbors x and y such that (ii) every vertex with f(v) = 1 has at least one neighbor assigned a 2 or 3 under f; (iii) the set of vertices assigned non-zero values under f is independent. The weight of an IDRDF is the sum of its values overs all vertices, and the independent double Roman domination number is the minimum weight of an IDRDF on In this article, we show that for every tree T of order where s(T) is the number of support vertices of T, improving the -upper bound established in [Maimani et al. Independent double Roman domination in graphs, Bulletin of the Iranian Mathematical Society, 46 (2020) 543–555]. Moreover, we characterize the trees T of order with

Double Roman Domination in Generalized Petersen Graphs P(ck, k)

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3}, satisfying the condition that every vertex u for which f(u)=1 is adjacent to at least one vertex assigned 2 or 3, and every vertex u with f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2. The weight of f equals the sum w(f)=∑v∈Vf(v). The minimum weight of a double Roman dominating function of G is called the double Roman domination number γdR(G) of a graph G. We obtain tight bounds and in some cases closed expressions for the double Roman domination number of generalized Petersen graphs P(ck,k). In short, we prove that γdR(P(ck,k))=32ck+ε, where limc→∞,k→∞εck=0.

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.

On the Double Roman Domination in Generalized Petersen Graphs P(5k,k)

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u)=1 is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k,k). It is proven that γdR(P(5k,k))=8k for k≡2,3mod5 and 8k≤γdR(P(5k,k))≤8k+2 for k≡0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k).

Double Roman domination subdivision number in graphs

For a graph [Formula: see text], a double Roman dominating function is a function [Formula: see text] having the property that if [Formula: see text], then vertex [Formula: see text] must have at least two neighbors assigned [Formula: see text] under [Formula: see text] or one neighbor with [Formula: see text], and if [Formula: see text], then vertex [Formula: see text] must have at least one neighbor with [Formula: see text]. The weight of a double Roman dominating function [Formula: see text] is the value [Formula: see text]. The double Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], equals the minimum weight of a double Roman dominating function on [Formula: see text]. The double Roman domination subdivision number [Formula: see text] of a graph [Formula: see text] is the minimum number of edges that must be subdivided (each edge in [Formula: see text] can be subdivided at most once) in order to increase the double Roman domination number. In this paper, we first show that the decision problem associated with sd[Formula: see text] is NP-hard and then establish upper bounds on the double Roman domination subdivision number for arbitrary graphs.