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Some Algorithmic Results on Restrained Domination in Graphs
[article]

2016
*
arXiv
*
pre-print

Finally, we give a new improved upper bound

arXiv:1606.02340v1
fatcat:lrg6szttbralxhuc2mi6aqsyka
*on*the*restrained**domination*number, cardinality of a minimum*restrained**dominating*set*in*terms of number of vertices and minimum degree of*graph*. ...*In*addition, we also propose polynomial time*algorithms*to solve the Minimum*Restrained**Domination*problem*in*cographs and chain*graphs*. ...*In*Section 2,*some*pertinent definitions and*some*preliminary*results*are discussed. ...##
###
NP-completeness and APX-completeness of restrained domination in graphs

2012
*
Theoretical Computer Science
*

*In*this paper, we investigate the NP-completeness of the

*restrained*

*domination*problem

*in*planar

*graphs*and split

*graphs*. ... Let G = (V , E) be a simple

*graph*. A vertex set S ⊆ V is a

*restrained*

*dominating*set if every vertex not

*in*S is adjacent to a vertex

*in*S and to a vertex

*in*V − S. ... Acknowledgments The first and the third authors are supported

*in*part by the National Natural Science Foundation of China ...

##
###
Split Restrained Dominating Set of an Interval Graph using an Algorithm

2012
*
International Journal of Applied Information Systems
*

There has been persistent

doi:10.5120/ijais12-450751
fatcat:viioqia72rbs3kqup3etpnagx4
*in*the*Algorithmic*aspects of interval*graphs**in*past decades spurred much by their numerous application of an interval*graphs*corresponding to an interval family I. A set ... The various applications of the theory of*Restrained**domination*, the most often discussed is communication network. ... This research was supported*in*part by B.Maheswari, Sri Padmavathi Women's University,Tirupati, Andhra Pradesh, Tirupati. ...##
###
On the complexity of the bondage and reinforcement problems

2012
*
Journal of Complexity
*

The bondage (resp. total bondage) number of a nonempty

doi:10.1016/j.jco.2011.11.001
fatcat:5pm3h3dzivexrpfgh6fzxwokpa
*graph*G is the smallest number of edges whose removal from G*results**in*a*graph*with larger*domination*(resp. total*domination*) number of G. ... The reinforcement (resp. total reinforcement) number of G is the smallest number of edges whose addition to G*results**in*a*graph*with smaller*domination*(resp. total*domination*) number. ... Acknowledgments The authors thank the anonymous referees for their kind comments and helpful suggestions*on*the original manuscript, which*resulted**in*this final version. ...##
###
Page 1555 of Mathematical Reviews Vol. , Issue 2002C
[page]

2002
*
Mathematical Reviews
*

A set S C V isa

*restrained**dominating*set if every vertex not*in*S is adjacent to a vertex*in*S and to a vertex*in*V — S. ... The*restrained**domination*number of G, denoted by y,(G), is the minimum cardinality of a*restrained**dominating*set of G. ...##
###
On Bondage Numbers of Graphs: A Survey with Some Comments

2013
*
International Journal of Combinatorics
*

The bondage number of a nonempty

doi:10.1155/2013/595210
fatcat:6zg7jz7i5jenzc2e2izfdy2txq
*graph*is the smallest number of edges whose removal from*results**in*a*graph*with*domination*number greater than the*domination*number of . ... This paper gives a survey*on*the bondage number, including known*results*, conjectures, problems, and*some*comments, also selectively summarizes other types of bondage numbers. ... the references [128] and [81] , which*resulted**in*this version. ...##
###
(1,j)-set problem in graphs
[article]

2014
*
arXiv
*
pre-print

*In*this paper, we first obtain an upper bound

*on*γ_(1,j) (G) using probabilistic methods, for bounded minimum and maximum degree

*graphs*. ... Finally, we design two

*algorithms*for finding γ_(1,j) (G) of a tree and a split

*graph*, for any fixed j, which answers an open question posed

*in*[CHHM13]. ... Using this

*result*, they also showed that

*domination*number is equal to

*restrained*

*domination*number. ...

##
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The restrained double Roman domination and graph operations
[article]

2021
*
arXiv
*
pre-print

And then we study the impact of

arXiv:2111.04363v1
fatcat:h4x6a2sblrcx7h57o3x7guh6x4
*some**graph*operations, such as strong product, cardinal product and corona with a*graph*,*on**restrained*double Roman*domination*number. ...*In*this paper, we first prove that the problem of computing RDRD-number is NP-hard even for chordal*graphs*. ... Strong ProductIn this section, we first present*some*bounds*on*the*restrained*double Roman*domination*number of strong product*graphs*, moreover we determine the exact values of*restrained*double Roman ...##
###
A note on restrained domination in self-complementary graphs

2019
*
Discussiones Mathematicae Graph Theory
*

The

doi:10.7151/dmgt.2222
fatcat:t6m5juc2cjczpftblay4obmeoy
*restrained**domination*number of a*graph*G is the minimum cardinality of a*restrained**dominating*set of G.*In*this paper, we study*restrained**domination**in*self-complementary*graphs*. ...*In*particular, we characterize the self-complementary*graphs*having equal*domination*and*restrained**domination*numbers. ... Acknowledgement Research of the 2nd and 3rd authors supported*in*part by the University of Johannesburg. ...##
###
Complexity of Bondage and Reinforcement
[article]

2011
*
arXiv
*
pre-print

The reinforcement number of G is the smallest number of edges whose addition to G

arXiv:1109.1657v1
fatcat:2nwepptkpnf7nddmw6aelmfu2q
*results**in*a*graph*with smaller*domination*number. ... The bondage (resp. total bondage) number of a nonempty*graph*G is the smallest number of edges whose removal from G*results**in*a*graph*with larger*domination*(resp. total*domination*) number of G. ... A survey of selected recent*results**on*total*domination**in*Henning [9] . ...##
###
Optimization Models for Efficient (t, r) Broadcast Domination in Graphs

2021
*
Symmetry
*

Known to be NP-complete,

doi:10.3390/sym13061028
fatcat:ouq32lajojcetjacry64c7lbsq
*domination*number problems*in**graphs*and networks arise*in*many real-life applications, ranging from the design of wireless sensor networks and biological networks to social networks ... While*some*theoretical approaches have been addressed for small values of t,r*in*the literature;*in*this work, we propose an approach from an optimization point of view. ... While a linear programming model has been applied to obtain*some**domination*parameters, e.g., strong*domination*,*restrained**domination*and strong*restrained**domination*[6] , weighted total*domination*...##
###
Restrained double Roman domination of a graph
[article]

2021
*
arXiv
*
pre-print

Then we present an upper bound

arXiv:2106.08501v1
fatcat:bhrlqt4lzffmbdlr7tpjebwfeu
*on*the*restrained*double Roman*domination*number of a connected*graph*G*in*terms of the order of G and characterize the*graphs*attaining this bound. ... The weight of a*restrained*double Roman*dominating*function f is the sum f(V)=∑_v∈Vf(v), and the minimum weight of a*restrained*double Roman*dominating*function*on*G is the*restrained*double Roman*domination*... RDRD number of*some**graphs**In*this section we investigate the exact value of the*restrained*double Roman*domination*number of*some**graphs*. Observation 3. ...##
###
On Bondage Numbers of Graphs -- a survey with some comments
[article]

2012
*
arXiv
*
pre-print

The bondage number of a nonempty

arXiv:1204.4010v1
fatcat:px2cuvbi6fhl5ci6hpoz5fyh3y
*graph*G is the cardinality of a smallest edge set whose removal from G*results**in*a*graph*with*domination*number greater than the*domination*number of G. ... This lecture gives a survey*on*the bondage number, including the known*results*, problems and conjectures. We also summarize other types of bondage numbers. ...*In*this section, we will state*some**results*and problems*on*the bondage number for a planar*graph*. Recall*some**results**on*planar*graphs*used*in*this section. ...##
###
γ-Excellent, critically dominated, end-dominated, and dot-critical trees are equivalent

2007
*
Discrete Mathematics
*

A

doi:10.1016/j.disc.2006.02.016
fatcat:k7hkttq6pvegremd6wn4nlqpie
*graph*is -excellent if every vertex of the*graph*is contained*in**some*minimum*dominating*set of the*graph*. A vertex v is critical*in*G if the*domination*number of G − v is smaller than that of G. ... The*graph*G is dot-critical if contracting any edge of G produces a*graph*with smaller*domination*number. G is critically*dominated*if the set of critical vertices forms a*dominating*set for G. ... Burton and Sumner [6] introduced the concept of a dot-critical*graph*as*one**in*which identifying any two adjacent vertices (i.e., contracting the edge comprising those vertices)*results**in*a*graph*with ...##
###
A note on trees, tables, and algorithms

2009
*
Networks
*

Several

doi:10.1002/net.20282
fatcat:z4oeob54b5dqxo6pg47lvv2bka
*algorithms*for optimal vertex subsets*in*trees are given by simple tables.*In*this paper we investigate the properties of and operations*on*these tables. ... We use known techniques for combining tables to correct a table*in*the literature; give a necessary and sufficient condition for a table to correspond to*some*tree property; discuss the question of dividing ... Figure 3 : 3 The tables for*restrained*-ness and*restrained**domination*. ...
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