The signed Roman domination number of two classes graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050024
Author(s):  
Xia Hong ◽  
Tianhu Yu ◽  
Zhengbang Zha ◽  
Huihui Zhang
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Double Star ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Closed Neighborhood

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. A signed Roman dominating function (SRDF) of [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) [Formula: see text] for each [Formula: see text], where [Formula: see text] is the set, called closed neighborhood of [Formula: see text], consists of [Formula: see text] and the vertex of [Formula: see text] adjacent to [Formula: see text] (ii) every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a SRDF [Formula: see text] is [Formula: see text]. The signed Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of a SRDF of [Formula: see text]. In this paper, we determine the exact values of signed Roman domination number of spider and double star. Specially, one of them generalizes the known result.

Quadruple Roman domination in graphs

2021 ◽  
pp. 2150130
Author(s):  
J. Amjadi ◽  
N. Khalili
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Graph Families ◽  
Domination In Graphs

Let [Formula: see text] be a finite and simple graph with vertex set [Formula: see text]. Let [Formula: see text] be a function that assigns label from the set [Formula: see text] to the vertices of a graph [Formula: see text]. For a vertex [Formula: see text], the active neighborhood of [Formula: see text], denoted by [Formula: see text], is the set of vertices [Formula: see text] such that [Formula: see text]. A quadruple Roman dominating function (QRDF) is a function [Formula: see text] satisfying the condition that for any vertex [Formula: see text] with [Formula: see text]. The weight of a QRDF is [Formula: see text]. The quadruple Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of a QRDF on [Formula: see text]. In this paper, we investigate the properties of the quadruple Roman domination number of graphs, present bounds on [Formula: see text] and give exact values for some graph families. In addition, complexity results are also obtained.

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.

The signed total Roman domatic number of a digraph

2018 ◽  
Vol 10 (02) ◽  
pp. 1850020 ◽  
Author(s):  
J. Amjadi
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Domatic Number ◽  
Vertex Set ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Dominating Function

Let [Formula: see text] be a finite simple digraph with vertex set [Formula: see text]. A signed total Roman dominating function (STRDF) on a digraph [Formula: see text] is a function [Formula: see text] such that (i) [Formula: see text] for every [Formula: see text], where [Formula: see text] consists of all inner neighbors of [Formula: see text], and (ii) every vertex [Formula: see text] for which [Formula: see text] has an inner neighbor [Formula: see text] for which [Formula: see text]. The weight of an STRDF [Formula: see text] is [Formula: see text]. The signed total Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of an STRDF on [Formula: see text]. A set [Formula: see text] of distinct STRDFs on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text] is called a signed total Roman dominating family (STRD family) (of functions) on [Formula: see text]. The maximum number of functions in an STRD family on [Formula: see text] is the signed total Roman domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we initiate the study of signed total Roman domatic number in digraphs and we present some sharp bounds for [Formula: see text]. In addition, we determine the signed total Roman domatic number of some classes of digraphs.

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.

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.

Trees with independent Roman domination number twice the independent domination number

2015 ◽  
Vol 07 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Mustapha Chellali ◽  
Nader Jafari Rad
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Independent Domination ◽  
Open Question ◽  
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] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a RDF [Formula: see text] is the value [Formula: see text]. The Roman domination number, [Formula: see text], of [Formula: see text] is the minimum weight of a RDF on [Formula: see text]. An RDF [Formula: see text] is called an independent Roman dominating function (IRDF) if the set [Formula: see text] is an independent set. The independent Roman domination number, [Formula: see text], is the minimum weight of an IRDF on [Formula: see text]. In this paper, we study trees with independent Roman domination number twice their independent domination number, answering an open question.

scholarly journals Signed strong Roman domination in graphs

2017 ◽  
Vol 48 (2) ◽  
pp. 135-147 ◽  
Author(s):  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar ◽  
Leila Asgharsharghi
Keyword(s):  
Maximum Degree ◽  
Domination Number ◽  
Simple Graph ◽  
Roman Domination ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Closed Neighborhood ◽  
Domination In Graphs ◽  
Dominating Function

Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximum degree $\Delta$. A signed strong Roman dominating function (abbreviated SStRDF) on a graph $G$ is a function $f:V\to \{-1,1,2,\ldots,\lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the conditions that (i) for every vertex $v$ of $G$, $\sum_{u\in N[v]} f(u)\ge 1$, where $N[v]$ is the closed neighborhood of $v$ and (ii) every vertex $v$ for which $f(v)=-1$ is adjacent to at least one vertex $u$ for which $f(u)\ge 1+\lceil\frac{1}{2}|N(u)\cap V_{-1}|\rceil$, where $V_{-1}=\{v\in V \mid f(v)=-1\}$. The minimum of the values $\sum_{v\in V} f(v)$, taken over all signed strong Roman dominating functions $f$ of $G$, is called the signed strong Roman domination number of $G$ and is denoted by $\gamma_{ssR}(G)$. In this paper we initiate the study of the signed strong Roman domination in graphs and present some (sharp) bounds for this parameter.

scholarly journals Some progress on the mixed Roman domination in graphs

2020 ◽  
Author(s):  
Hossein Abdollahzadeh Ahangar ◽  
Jafar Amjadi ◽  
Mustapha Chellali ◽  
S. Kosari ◽  
Vladimir Samodivkin ◽  
...  
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Np Complete ◽  
Domination In Graphs ◽  
Dominating Function

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. A mixed Roman dominating function (MRDF) of $G$ is a function $f:V\cup E\rightarrow \{0,1,2\}$ satisfying the condition that every element $x\in V\cup E$ for which $f(x)=0$ is adjacent or incident to at least one element $% y\in V\cup E$ for which $f(y)=2$. The weight of a mixed Roman dominating function $f$ is $\omega (f)=\sum_{x\in V\cup E}f(x)$. The mixed Roman domination number $\gamma _{R}^{\ast }(G)$ of $G$ is the minimum weight of a mixed Roman dominating function of $G$. We first show that the problem of computing $\gamma _{R}^{\ast }(G)$ is NP-complete for bipartite graphs and then we present upper and lower bounds on the mixed Roman domination number, some of them are for the class of trees.

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.

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