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.

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.

Independent Roman domination and 2-independence in trees

2018 ◽  
Vol 10 (04) ◽  
pp. 1850052
Author(s):  
J. Amjadi ◽  
S. M. Sheikholeslami ◽  
M. Valinavaz ◽  
N. Dehgardi
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Two Parameters ◽  
Dominating Function

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. A Roman dominating function 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]. A Roman dominating function [Formula: see text] is called an independent Roman dominating function if the set of all vertices with positive weights is an independent set. The weight of an independent Roman dominating function [Formula: see text] is the value [Formula: see text]. The independent Roman domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an independent Roman dominating function on [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a 2-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 2-independent set of [Formula: see text] is the 2-independence number [Formula: see text]. These two parameters are incomparable in general, however, we show that for any tree [Formula: see text], [Formula: see text] and we characterize all trees attaining the equality.

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.

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.

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.

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 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.

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.

Graphs whose weak Roman domination number increases by the deletion of any edge

2021 ◽  
Author(s):  
Rihab Hamid ◽  
Nour El Houda Bendahib ◽  
Mustapha Chellali ◽  
Nacéra Meddah
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Necessary Condition ◽  
Roman Domination ◽  
Edge Deletion ◽  
Critical Graphs ◽  
Roman Domination Number ◽  
Number Formula ◽  
Roman Dominating Function

Let [Formula: see text] be a function on a graph [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is said to be undefended with respect to [Formula: see text] if it is not adjacent to a vertex [Formula: see text] with [Formula: see text]. A function [Formula: see text] is called a weak Roman dominating function (WRDF) if each vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text], such that the function [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text], has no undefended vertex. The weight of a WRDF is the sum of its function values over all vertices, and the weak Roman domination number [Formula: see text] is the minimum weight of a WRDF in [Formula: see text]. In this paper, we consider the effects of edge deletion on the weak Roman domination number of a graph. We show that the deletion of an edge of [Formula: see text] can increase the weak Roman domination number by at most 1. Then we give a necessary condition for [Formula: see text]-ER-critical graphs, that is, graphs [Formula: see text] whose weak Roman domination number increases by the deletion of any edge. Restricted to the class of trees, we provide a constructive characterization of all [Formula: see text]-ER-critical trees.

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.

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