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Irredundance and domination in kings graphs

2003
*
Discrete Mathematics
*

A set of

doi:10.1016/s0012-365x(02)00494-6
fatcat:swz7jhnrxbckhopkqqth46sco4
*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] . ...##
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There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations ofK14

2009
*
Journal of combinatorial designs (Print)
*

Consider a number of

doi:10.1002/jcd.20188
fatcat:gqreaywcwnacjhodu5oebzzn64
*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 ...##
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Breaking the 2n-barrier for Irredundance: Two lines of attack

2011
*
Journal of Discrete Algorithms
*

The lower

doi:10.1016/j.jda.2011.03.002
fatcat:txstt3rsnbbcdppraf4sqyxm6q
*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

2018
*
Annals of Pure and Applied Mathematics
*

Various studies had been done on these chessboard problems

doi:10.22457/apam.v18n1a8
fatcat:5si26pymojdc7l43tcwge3mc3q
*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]

2010
*
Lecture Notes in Computer Science
*

The lower

doi:10.1007/978-3-642-13073-1_28
fatcat:qxiifrdksfbqle7blmkdwd7yc4
*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]

2009
*
arXiv
*
pre-print

The lower

arXiv:0909.4224v1
fatcat:bvjx4me43zgt5ddhvc7bsaqwcq
*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] . ...##
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Irredundant and perfect neighborhood sets in graphs and claw-free graphs

1999
*
Discrete Mathematics
*

of a

doi:10.1016/s0012-365x(98)00239-8
fatcat:f4jll2sbznepfced7uca6syiye
*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~. ...##
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Irredundant and perfect neighborhood sets in graphs and claw-free graphs

1999
*
Discrete Mathematics
*

of a

doi:10.1016/s0012-365x(99)90073-0
fatcat:lkthv24v45cu7cnrdp5j5bi6iq
*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

doi:10.1016/s0012-365x(05)00575-3
fatcat:rlzgd6adf5gy3hfmpg6dmxfmmm
*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

doi:10.1016/s0012-365x(02)00856-7
fatcat:sdrhosxi6bhpxkg534tmkzygei
*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

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. ...

##
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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

doi:10.1016/s0012-365x(98)90328-4
fatcat:s2tsivncvfcilf6jlbl4fx24zq
*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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