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### Irredundance and domination in kings graphs

Odile Favaron, Gerd H. Fricke, Dan Pritikin, Joël Puech
2003 Discrete Mathematics
A set of kings is said to form an irredundant set if each attacks a square attacked by no other king in the set.  ...  We prove that the maximum size of an irredundant set of kings is bounded between (n − 1) 2 =3 and n 2 =3, and that the minimum size of a maximal irredundant set of kings is bounded between n 2 =9 and (  ...  ) in kings graphs, thereby answering Problem K.5.3 of [3] .  ...

### There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations ofK14

Petteri Kaski, Patric R. J. Östergård
2009 Journal of combinatorial designs (Print)
Consider a number of kings we want to place on the vertices our graph (the irredundant set vertices).  ...  For example, a king has no right of existence if all its neighbouring vertices contain a king, or if has one neighbouring king (which puts his own vertex in dispute) and all other neighbouring vertices  ...

### Page 789 of Mathematical Reviews Vol. , Issue 2004b [page]

2004 Mathematical Reviews
H. (1-WRTS; Dayton, OH); Pritikin, Dan (1- MMOH; Oxford, OH); Puech, Joél (F-PARIS11-RI; Orsay) Irredundance and domination in kings graphs.  ...  The kings graph, denoted K,, in this paper, is the graph whose vertex set consists of the squares of an n by n chessboard, where two vertices are adjacent if and only if a king can move from one square  ...

### Breaking the 2n-barrier for Irredundance: Two lines of attack

Daniel Binkele-Raible, Ljiljana Brankovic, Marek Cygan, Henning Fernau, Joachim Kneis, Dieter Kratsch, Alexander Langer, Mathieu Liedloff, Marcin Pilipczuk, Peter Rossmanith, Jakub Onufry Wojtaszczyk
2011 Journal of Discrete Algorithms
The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G), respectively, are conceptually linked to the domination and independence numbers and have numerous relations to other  ...  graph parameters.  ...  In fact, a set is minimal dominating if and only if it is irredundant and dominating [13] .  ...

### Perfect Domination for Bishops, Kings and Rooks Graphs On Square Chessboard

K.S.P. Sowndarya, Sri Sathya Sai Institute of Higher Learning, Y. Lakshmi Naidu
2018 Annals of Pure and Applied Mathematics
Various studies had been done on these chessboard problems in relation to different domination parameters such as domination, independence and irredundance on queens, bishops, kings and rooks graphs.  ...  In this paper we extend this study on perfect domination and determine the exact values of Perfect Domination number for Bishops graph B n , Kings Graph K n , and Rooks Graph R n on an n × n chessboard  ...  We pay our sincere thanks to all the authors, professors, and experts for their contributions, and also would like to thank the reviwers for their useful suggestions.  ...

### A Parameterized Route to Exact Puzzles: Breaking the 2 n -Barrier for Irredundance [chapter]

Daniel Binkele-Raible, Ljiljana Brankovic, Henning Fernau, Joachim Kneis, Dieter Kratsch, Alexander Langer, Mathieu Liedloff, Peter Rossmanith
2010 Lecture Notes in Computer Science
The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G) respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph  ...  It is a long-standing open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time less than the trivial Ω(2 n ) enumeration barrier.  ...  In fact, a set is minimal dominating if and only if it is irredundant and dominating [7] .  ...

### Breaking the 2^n-Barrier for Irredundance: A Parameterized Route to Solving Exact Puzzles [article]

Ljiljana Brankovic, Henning Fernau, Joachim Kneis, Dieter Kratsch Alexander Langer Mathieu Liedloff Daniel Raible Peter Rossmanith
2009 arXiv   pre-print
The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G) respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph  ...  It is a long-standing open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time less than the trivial Ω(2^n) enumeration barrier.  ...  In fact, a set is minimal dominating if and only if it is irredundant and dominating [7] .  ...

### Irredundant and perfect neighborhood sets in graphs and claw-free graphs

O Favaron
1999 Discrete Mathematics
of a graph G.  ...  Let O(G),O~(G),ir(G),sir(G) be the minimum cardinality of, respectively, a perfect neighborhood set, an independent perfect neighborhood set, a maximal irredundant set and a semimaximal irredundant set  ...  As noticed in [2] , ever¢ PN-set X is irredundant since every vertex of y~ must be dominated by B~.  ...

### Irredundant and perfect neighborhood sets in graphs and claw-free graphs

Odile Favaron, Joël Puech
1999 Discrete Mathematics
of a graph G.  ...  Let O(G),O~(G),ir(G),sir(G) be the minimum cardinality of, respectively, a perfect neighborhood set, an independent perfect neighborhood set, a maximal irredundant set and a semimaximal irredundant set  ...  As noticed in [2] , ever¢ PN-set X is irredundant since every vertex of y~ must be dominated by B~.  ...

### Author index to volume 305

2005 Discrete Mathematics
Volkmann and I. Zverovich, Unique irredundance, domination and independent domination in graphs J.H., see G.S. Domke (1-3) 112-122 Haynes, T.W., see R.C. Brigham (1-3) 18-32 He, W., see Y.  ...  Markus, On weakly connected domination in graphs II (1-3) 112-122 Dougherty, S.T., S.Y. Kim and Y.H.  ...

### Contents

2003 Discrete Mathematics
Puech Irredundance and domination in kings graphs 131 H.A. Harutyunyan and A.L. Liestman On the monotonicity of the broadcast function 149 T.W. Haynes, S.T. Hedetniemi, M.A. Henning and P.J.  ...  Mollard On paths and cycles dominating hypercubes 121 C.D. Savage, I. Shields and D.B. West On the existence of Hamiltonian paths in the cover graph of MðnÞ 241 K. Betsumiya, S. Georgiou, T.A.  ...

### α-Domination

J.E. Dunbar, D.G. Hoffman, R.C. Laskar, L.R. Markus
2000 Discrete Mathematics
In this paper, we introduce -domination, discuss bounds for 1=2 (G) for the King's graph, and give bounds for (G) for a general , 0 ¡ 61.  ...  Let G =(V; E) be any graph with n vertices, m edges and no isolated vertices. For some with 0 ¡ 61 and a set S The size of a smallest such S is called the -domination number and is denoted by (G).  ...  Woodall and the anonymous referees for their many helpful suggestions leading to the present form of this paper.  ...

### Page 6200 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews
Rédl [in Graphs, hypergraphs and block systems (Zielona Gora, 1976), 2\1- 219, College Engrg., Zielona Gora, 1976; Zbl 337:05133}: Given any acyclic digraph D, there exists a graph G with G “ D.  ...  Graph Theory 1 (1977), no. 3, 227-268; MR 58 #372] and the update by Bondy [in Surveys in combinatorics, 1991 (Guild- ford, 1991), 221-252, Cambridge Univ. Press, Cambridge, 1991; MR 93¢e:05071].  ...

### Page 6584 of Mathematical Reviews Vol. , Issue 97K [page]

1997 Mathematical Reviews
Mynhardt, Domination and irredundance in cubic graphs (205-214); Peter Cowling, The total graph of a hyper- graph (215-236); Gayla S. Domke, Jean E. Dunbar and Lisa R.  ...  Hedetniemi and Alice A. McRae, On weakly connected domination in graphs (261-269); Jonathan David Farley, Perfect sequences of chain-complete posets (271-296); M. A. Fiol, E. Garriga and J. L. A.  ...

### Master index of volumes 181–190

1998 Discrete Mathematics
Liu and B. Xu, On endo-homology of complexes of graphs (Note) Huang, Kings in quasi-transitive digraphs 185 (1998) Barcucci, E., S. Brunetti, A. Del Lungo and F.  ...  Zverovich, Upper domination and upper irredundance perfect graphs 190 (1998) Gutin, G., A note on the cardinality of certain classes of unlabeled multipartite tournaments (Communication) 186 (1998  ...
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