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.

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 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.

Critical concept for double Roman domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050020
Author(s):  
S. Nazari-Moghaddam ◽  
L. Volkmann
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Critical Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Domination In Graphs ◽  
Dominating Function

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

2018 ◽  
Author(s):  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao ◽  
Jia-Bao Liu
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Generalized Petersen Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Petersen Graphs ◽  
Roman Dominating Function ◽  
Dominating Function ◽  
Domination Numbers

A double Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2, 3} 2 with the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for 3 which f(v) = 3 or two vertices v1 and v2 for which f(v1) = f(v2) = 2, and every vertex u for which 4 f(u) = 1 is adjacent to at least one vertex v for which f(v) ≥ 2. The weight of a double Roman dominating function f is the value w(f) = ∑u∈V(G) 5 f(u). The minimum weight over all double 6 Roman dominating functions on a graph G is called the double Roman domination number γdR(G) 7 of G. In this paper we determine the exact value of the double Roman domination number of the 8 generalized Petersen graphs P(n, 2) by using a discharging approach.

Double Roman domination subdivision number in graphs

2021 ◽  
Author(s):  
J. Amjadi ◽  
H. Sadeghi
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Minimum Number ◽  
Roman Dominating Function ◽  
Arbitrary Graphs ◽  
Domination Subdivision Number ◽  
Dominating Function

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.

A linear algorithm for double Roman domination of proper interval graphs

2019 ◽  
Vol 12 (01) ◽  
pp. 2050011 ◽  
Author(s):  
Abolfazl Poureidi
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Interval Graph ◽  
Roman Domination ◽  
Linear Algorithm ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Proper Interval Graph ◽  
Roman Dominating Function ◽  
Dominating Function

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.

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

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 119
Author(s):  
Darja Rupnik Poklukar ◽  
Janez Žerovnik
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Upper Bounds ◽  
Roman Domination ◽  
Generalized Petersen Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Petersen Graphs ◽  
Roman Dominating Function ◽  
Dominating Function

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).

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