scholarly journals ON THE ROMAN BONDAGE NUMBER OF A GRAPH

2013 ◽  
Vol 05 (01) ◽  
pp. 1350001 ◽  
Author(s):  
A. BAHREMANDPOUR ◽  
FU-TAO HU ◽  
S. M. SHEIKHOLESLAMI ◽  
JUN-MING XU
Keyword(s):  
Decision Problem ◽  
Maximum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Np Hard ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Bondage Number ◽  
Roman Domination Number ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a RDF is the value f(V(G)) = Σu∈V(G) f(u). The minimum weight of a RDF on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number bR(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E′ ⊆ E(G) for which γR(G - E′) > γR(G). In this paper, we first show that the decision problem for determining bR(G) is NP-hard even for bipartite graphs and then we establish some sharp bounds for bR(G) and characterizes all graphs attaining some of these bounds.

scholarly journals The Roman bondage number of a digraph

2016 ◽  
Vol 47 (4) ◽  
pp. 421-431
Author(s):  
Seyed Mahmoud Sheikholeslami ◽  
Nasrin Dehgardi ◽  
Lutz Volkmann ◽  
Dirk Meierling
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Bondage Number ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Dominating Function

Let $D=(V,A)$ be a finite and simple digraph. A  Roman dominating function on $D$ is a labeling $f:V (D)\rightarrow \{0, 1, 2\}$ such that every vertex with label 0 has an in-neighbor with label 2. The weight of an RDF $f$ is the value $\omega(f)=\sum_{v\in V}f (v)$. The minimum weight of a Roman dominating function on a digraph $D$ is called the Roman domination number, denoted by $\gamma_{R}(D)$. The Roman bondage number $b_{R}(D)$ of a digraph $D$ with maximum out-degree at least two is the minimum cardinality of all sets $A'\subseteq A$ for which $\gamma_{R}(D-A')>\gamma_R(D)$. In this paper, we initiate the study of the Roman bondage number of a digraph. We determine the Roman bondage number in several classes of digraphs and give some sharp bounds.

scholarly journals On the double Roman bondage numbers of graphs

2022 ◽  
Author(s):  
N. Jafari Rad ◽  
H. R. Maimani ◽  
M. Momeni ◽  
F. Rahimi Mahid
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Bondage Number ◽  
Roman Domination Number ◽  
Complexity Issue ◽  
Double Roman Domination ◽  
Sum Formula ◽  
Roman Dominating Function

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.

Roman domination in graphs: The class ℛUV R

2016 ◽  
Vol 08 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Vladimir Samodivkin
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Sufficient Conditions ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Roman Domination Number ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Necessary And Sufficient ◽  
Domination In Graphs

For a graph [Formula: see text], a Roman dominating function (RDF) [Formula: see text] has the property that every vertex [Formula: see text] with [Formula: see text] has a neighbor [Formula: see text] with [Formula: see text]. The weight of a RDF [Formula: see text] is the sum [Formula: see text], and the minimum weight of a RDF on [Formula: see text] is the Roman domination number [Formula: see text] of [Formula: see text]. The Roman bondage number [Formula: see text] of [Formula: see text] is the minimum cardinality of all sets [Formula: see text] for which [Formula: see text]. A graph [Formula: see text] is in the class [Formula: see text] if the Roman domination number remains unchanged when a vertex is deleted. In this paper, we obtain tight upper bounds for [Formula: see text] and [Formula: see text] provided a graph [Formula: see text] is in [Formula: see text]. We present necessary and sufficient conditions for a tree to be in the class [Formula: see text]. We give a constructive characterization of [Formula: see text]-trees using labelings.

scholarly journals Strong equality of Roman and perfect Roman domination in trees

2022 ◽  
Author(s):  
Zehui Shao ◽  
Saeed Kosari ◽  
Hadi Rahbani ◽  
Mehdi Sharifzadeh ◽  
Seyed Mahmoud Sheikholeslami
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Np Hard ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Dominating Function

A Roman dominating function (RD-function) on a graph $G = (V, E)$ is a function $f: V \longrightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. An Roman dominating function $f$ in a graph $G$ is perfect Roman dominating function (PRD-function) if  every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex  $v$ for which $f(v) = 2$. The (perfect) Roman domination number $\gamma_R(G)$ ($\gamma_{R}^{p}(G)$) is the minimum weight of an (perfect) Roman dominating function on $G$.  We say that $\gamma_{R}^{p}(G)$ strongly equals $\gamma_R(G)$, denoted by $\gamma_{R}^{p}(G)\equiv \gamma_R(G)$, if every RD-function on $G$ of minimum weight is a PRD-function. In this paper we  show that for a given graph $G$, it is NP-hard to decide whether $\gamma_{R}^{p}(G)= \gamma_R(G)$ and also we provide a constructive characterization of trees $T$ with $\gamma_{R}^{p}(T)\equiv \gamma_R(T)$.

scholarly journals Total Weak Roman Domination in Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 831 ◽  
Author(s):  
Abel Cabrera Martínez ◽  
Luis P. Montejano ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Open Neighbourhood ◽  
Np Hard ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Close Relationship ◽  
Roman Dominating Function ◽  
Domination In Graphs ◽  
Dominating Function

Given a graph G = ( V , E ) , a function f : V → { 0 , 1 , 2 , ⋯ } is said to be a total dominating function if ∑ u ∈ N ( v ) f ( u ) > 0 for every v ∈ V , where N ( v ) denotes the open neighbourhood of v. Let V i = { x ∈ V : f ( x ) = i } . We say that a function f : V → { 0 , 1 , 2 } is a total weak Roman dominating function if f is a total dominating function and for every vertex v ∈ V 0 there exists u ∈ N ( v ) ∩ ( V 1 ∪ V 2 ) such that the function f ′ , defined by f ′ ( v ) = 1 , f ′ ( u ) = f ( u ) - 1 and f ′ ( x ) = f ( x ) whenever x ∈ V ∖ { u , v } , is a total dominating function as well. The weight of a function f is defined to be w ( f ) = ∑ v ∈ V f ( v ) . In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by γ t r ( G ) , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on γ t r ( G ) and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.

scholarly journals On a Relation between the Perfect Roman Domination and Perfect Domination Numbers of a Tree

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 966
Author(s):  
Zehui Shao ◽  
Saeed Kosari ◽  
Mustapha Chellali ◽  
Seyed Mahmoud Sheikholeslami ◽  
Marzieh Soroudi
Keyword(s):  
Dominating Set ◽  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Dominating Function ◽  
Domination Numbers

A dominating set in a graph G is a set of vertices S ⊆ V ( G ) such that any vertex of V − S is adjacent to at least one vertex of S . A dominating set S of G is said to be a perfect dominating set if each vertex in V − S is adjacent to exactly one vertex in S. The minimum cardinality of a perfect dominating set is the perfect domination number γ p ( G ) . A function f : V ( G ) → { 0 , 1 , 2 } is a perfect Roman dominating function (PRDF) on G if every vertex u ∈ V for which f ( u ) = 0 is adjacent to exactly one vertex v for which f ( v ) = 2 . The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of G is the perfect Roman domination number γ R p ( G ) . In this paper, we prove that for any nontrivial tree T, γ R p ( T ) ≥ γ p ( T ) + 1 and we characterize all trees attaining this bound.

scholarly journals Quadruple Roman Domination in Trees

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1318
Author(s):  
Zheng Kou ◽  
Saeed Kosari ◽  
Guoliang Hao ◽  
Jafar Amjadi ◽  
Nesa Khalili
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Theoretical Computer Science ◽  
Upper And Lower Bounds ◽  
Discrete Mathematics ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function

This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)<k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

scholarly journals Double Roman Graphs in P(3k, k)

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 336
Author(s):  
Zehui Shao ◽  
Rija Erveš ◽  
Huiqin Jiang ◽  
Aljoša Peperko ◽  
Pu Wu ◽  
...  
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Generalized Petersen Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Petersen Graphs ◽  
Roman Dominating Function ◽  
Sharp Lower Bound ◽  
Dominating Function

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} with the properties that if f(u)=0, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f(u)=1, then vertex u 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 is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γdR(G)=3γ(G), where γ(G) is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs P(3k,k), and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs P(3k,k). This implies that P(3k,k) is a double Roman graph if and only if either k≡0 (mod 3) or k∈{1,4}.

Roman domination and 2-independence in trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Nacéra Meddah ◽  
Mustapha Chellali
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Sum Formula ◽  
Roman Dominating Function

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] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-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 [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

Resolving Roman domination in graphs

2021 ◽  
Author(s):  
P. Roushini Leely Pushpam ◽  
B. Mahavir ◽  
M. Kamalam
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Domination In Graphs ◽  
Dominating Function

Let [Formula: see text] be a graph and [Formula: see text] be a Roman dominating function defined on [Formula: see text]. Let [Formula: see text] be some ordering of the vertices of [Formula: see text]. For any [Formula: see text], [Formula: see text] is defined by [Formula: see text]. If for all [Formula: see text], [Formula: see text], we have [Formula: see text], that is [Formula: see text], for some [Formula: see text], then [Formula: see text] is called a resolving Roman dominating function (RDF) on [Formula: see text]. The weight of a resolving RDF [Formula: see text] on [Formula: see text] is [Formula: see text]. The minimum weight of a resolving RDF on [Formula: see text] is called the resolving Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. A resolving RDF on [Formula: see text] with weight [Formula: see text] is called a [Formula: see text]-function on [Formula: see text]. In this paper, we find the resolving Roman domination number of certain well-known classes of graphs. We also categorize the class of graphs whose resolving Roman domination number equals their order.

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