scholarly journals The double Roman domination number of generalized Sierpiński graphs

2020 ◽  
Vol 12 (04) ◽  
pp. 2050047
Author(s):  
V. Anu ◽  
S. Aparna Lakshmanan
Keyword(s):  
Domination Number ◽  
Roman Domination ◽  
Sierpiński Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination

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].

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}.

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.

An upper bound on the double Roman domination number

2018 ◽  
Vol 36 (1) ◽  
pp. 81-89 ◽  
Author(s):  
J. Amjadi ◽  
S. Nazari-Moghaddam ◽  
S. M. Sheikholeslami ◽  
L. Volkmann
Keyword(s):  
Upper Bound ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Double Roman Domination

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

2019 ◽  
Vol 270 ◽  
pp. 159-167 ◽  
Author(s):  
Rana Khoeilar ◽  
Hossein Karami ◽  
Mustapha Chellali ◽  
Seyed Mahmoud Sheikholeslami
Keyword(s):  
Upper Bound ◽  
Minimum Degree ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Double Roman Domination

scholarly journals Bounds on the total double Roman domination number of graphs

2021 ◽  
Author(s):  
Guoliang Hao ◽  
Zhihong Xie ◽  
Seyed Mahmoud Sheikholeslami ◽  
M. Hajjari
Keyword(s):  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Double Roman Domination

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.

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.

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.

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