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).

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}.

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

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 206 ◽  
Author(s):  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao ◽  
Yongsheng Rao ◽  
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 (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.

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.

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 Outer-Independent Double Roman Domination of Graphs

2021 ◽  
Vol 6 ◽  
Author(s):  
Yongsheng Rao ◽  
Saeed Kosari ◽  
Seyed Mahmoud Sheikholeslami ◽  
M. Chellali ◽  
Mahla Kheibari
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Roman Dominating Function ◽  
Dominating Function

An outer-independent double Roman dominating function (OIDRDF) of a graph G is a function h:V(G)→{0,1,2,3} such that i) every vertex v with f(v)=0 is adjacent to at least one vertex with label 3 or to at least two vertices with label 2, ii) every vertex v with f(v)=1 is adjacent to at least one vertex with label greater than 1, and iii) all vertices labeled by 0 are an independent set. The weight of an OIDRDF is the sum of its function values over all vertices. The outer-independent double Roman domination number γoidR (G) is the minimum weight of an OIDRDF on G. It has been shown that for any tree T of order n ≥ 3, γoidR (T) ≤ 5n/4 and the problem of characterizing those trees attaining equality was raised. In this article, we solve this problem and we give additional bounds on the outer-independent double Roman domination number. In particular, we show that, for any connected graph G of order n with minimum degree at least two in which the set of vertices with degree at least three is independent, γoidR (T) ≤ 4n/3.

scholarly journals Some Properties of Double Roman Domination

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Hong Yang ◽  
Xiaoqing Zhou
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Roman Dominating Function ◽  
Domination In Graphs ◽  
Dominating Function

A double Roman dominating function on a graph G is a function f:VG⟶0,1,2,3 satisfying the conditions that every vertex u for which fu=0 is adjacent to at least one vertex v for which fv=3 or two vertices v1 and v2 for which fv1=fv2=2 and every vertex u for which fu=1 is adjacent to at least one vertex v for which fv≥2. The weight of a double Roman dominating function f is the value fV=∑u∈Vfu. The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination numberγdRG of G. A graph with γdRG=3γG is called a double Roman graph. In this paper, we study properties of double Roman domination in graphs. Moreover, we find a class of double Roman graphs and give characterizations of trees with γdRT=γRT+k for k=1,2.

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.

Twin signed total Roman domination numbers in digraphs

2018 ◽  
Vol 11 (03) ◽  
pp. 1850034 ◽  
Author(s):  
J. Amjadi ◽  
M. Soroudi
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Dominating Function ◽  
Domination Numbers

Let [Formula: see text] be a finite simple digraph with vertex set [Formula: see text] and arc set [Formula: see text]. A twin signed total Roman dominating function (TSTRDF) on the digraph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) [Formula: see text] and [Formula: see text] for each [Formula: see text], where [Formula: see text] (respectively [Formula: see text]) consists of all in-neighbors (respectively out-neighbors) of [Formula: see text], and (ii) every vertex [Formula: see text] for which [Formula: see text] has an in-neighbor [Formula: see text] and an out-neighbor [Formula: see text] with [Formula: see text]. The weight of an TSTRDF [Formula: see text] is [Formula: see text]. The twin signed total Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of an TSTRDF on [Formula: see text]. In this paper, we initiate the study of twin signed total Roman domination in digraphs and we present some sharp bounds on [Formula: see text]. In addition, we determine the twin signed Roman domination number of some classes of digraphs.

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