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Some Algorithmic Results on Restrained Domination in Graphs [article]

Arti Pandey, B.S. Panda
2016 arXiv   pre-print
Finally, we give a new improved upper bound 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.  ... 
arXiv:1606.02340v1 fatcat:lrg6szttbralxhuc2mi6aqsyka

NP-completeness and APX-completeness of restrained domination in graphs

Lei Chen, Weiming Zeng, Changhong Lu
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  ... 
doi:10.1016/j.tcs.2012.05.005 fatcat:ejl2wswvufd7xhdztundoushgu

Split Restrained Dominating Set of an Interval Graph using an Algorithm

A. Sudhakaraiah, E. Gnana Deepika, V. Ramalatha
2012 International Journal of Applied Information Systems  
There has been persistent 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.  ... 
doi:10.5120/ijais12-450751 fatcat:viioqia72rbs3kqup3etpnagx4

On the complexity of the bondage and reinforcement problems

Fu-Tao Hu, Jun-Ming Xu
2012 Journal of Complexity  
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.  ...  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.  ... 
doi:10.1016/j.jco.2011.11.001 fatcat:5pm3h3dzivexrpfgh6fzxwokpa

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

Jun-Ming Xu
2013 International Journal of Combinatorics  
The bondage number of a nonempty 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.  ... 
doi:10.1155/2013/595210 fatcat:6zg7jz7i5jenzc2e2izfdy2txq

(1,j)-set problem in graphs [article]

Arijit Bishnu, Kunal Dutta, Arijit Ghosh, Subhabrata Paul
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.  ... 
arXiv:1410.3091v1 fatcat:47ow2xoapbgbjjle4nswjgcrqe

The restrained double Roman domination and graph operations [article]

Zhipeng Gao, Changqing Xi, Jun Yue
2021 arXiv   pre-print
And then we study the impact of 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  ... 
arXiv:2111.04363v1 fatcat:h4x6a2sblrcx7h57o3x7guh6x4

A note on restrained domination in self-complementary graphs

Wyatt J. Desormeaux, Teresa W. Haynes, Michael .A. Henning
2019 Discussiones Mathematicae Graph Theory  
The 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.  ... 
doi:10.7151/dmgt.2222 fatcat:t6m5juc2cjczpftblay4obmeoy

Complexity of Bondage and Reinforcement [article]

Fu-Tao Hu, Jun-Ming Xu
2011 arXiv   pre-print
The reinforcement number of G is the smallest number of edges whose addition to G 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] .  ... 
arXiv:1109.1657v1 fatcat:2nwepptkpnf7nddmw6aelmfu2q

Optimization Models for Efficient (t, r) Broadcast Domination in Graphs

Poompol Buathong, Tipaluck Krityakierne
2021 Symmetry  
Known to be NP-complete, 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  ... 
doi:10.3390/sym13061028 fatcat:ouq32lajojcetjacry64c7lbsq

Restrained double Roman domination of a graph [article]

Doost Ali Mojdeh, Iman Masoumi, Lutz Volkmann
2021 arXiv   pre-print
Then we present an upper bound 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.  ... 
arXiv:2106.08501v1 fatcat:bhrlqt4lzffmbdlr7tpjebwfeu

On Bondage Numbers of Graphs -- a survey with some comments [article]

Jun-Ming Xu
2012 arXiv   pre-print
The bondage number of a nonempty 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.  ... 
arXiv:1204.4010v1 fatcat:px2cuvbi6fhl5ci6hpoz5fyh3y

γ-Excellent, critically dominated, end-dominated, and dot-critical trees are equivalent

Tamara Burton, David P. Sumner
2007 Discrete Mathematics  
A 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  ... 
doi:10.1016/j.disc.2006.02.016 fatcat:k7hkttq6pvegremd6wn4nlqpie

A note on trees, tables, and algorithms

Wayne Goddard, Stephen T. Hedetniemi
2009 Networks  
Several 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.  ... 
doi:10.1002/net.20282 fatcat:z4oeob54b5dqxo6pg47lvv2bka
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