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Tiling a simply connected figure with bars of length 2 or 3

1996
*
Discrete Mathematics
*

We first prove that if F has no peak (

doi:10.1016/0012-365x(96)00158-6
fatcat:bgh3kt443zbbfhayoaocgwwo7a
*a*peak is*a*cell*of*F which has three*of*its edges in the contour*of*F), then F can be*tiled**with*rectangular*bars*formed from*2**or**3*cells. ... Let F be*a**simply**connected**figure*formed from*a*finite set*of*cells*of*the planar square lattice. ...*of**tiling**with**2*-*bars*and*3*-*bars*: the input is*a**simply**connected*fgure F and the algorithm tells whether F can be*tiled**with**2*-*bars*and*3*-*bars**or*not. ...##
###
Tiling with Small Tiles
[article]

2015
*
arXiv
*
pre-print

We also present

arXiv:1511.03043v1
fatcat:ar4gbtt6pzbwzoctatw4zg6lta
*a*result to*a*more classic*tiling*question*with*dominoes and L-shape*tiles*. ... We look at sets*of**tiles*that can*tile*any region*of*size greater than 1 on the square grid. ...*Figure**3*.*3*Local move 1*Figure*4 . 4 Local move*2**Figure*5 . 5 Set S*2*:*a*domino, L-*tile*,*3*-*bar*, T-*tile*, and plus-*tile*Lemma*3*.*2*. 32 The set S*2*given inFigure 5is*a*fountain set.Proof. ...##
###
Page 5341 of Mathematical Reviews Vol. , Issue 97I
[page]

1997
*
Mathematical Reviews
*

{For the entire collection see MR 97h:05003.}
971:05025 05B45 52C20
Rémila, Eric

*Tiling**a**simply**connected**figure**with**bars**of**length**2**or**3*. ... We first prove that if F has no peak (*a*peak is*a*cell*of*F which has three*of*its edges in the contour*of*F), then F can be*tiled**with*rectangular*bars*formed from*2**or**3*cells. ...##
###
Tiling groups: New applications in the triangular lattice

1998
*
Discrete & Computational Geometry
*

Afterward, we use

doi:10.1007/pl00009382
fatcat:6oyff6wblfbpjpz7ikhwfn7qti
*a*similar method to get*a*linear algorithm to*tile*polygons*with*m-leaning*bars*(parallelograms*of**length*m formed from 2m cells*of**A*) and equilateral triangles (whose sides have*length*... m) and we produce*a*quadratic algorithm to*tile*polygons*with*m-leaning*bars*. ... Kenyon [KK] , which obtained an algorithm to*tile**a**simply**connected**figure**with*m-*bars*(rectangles*of**length*m and unit width). In Section*2*we recall the notions introduced by J. H. Conway, J. C. ...##
###
A note on tiling with integer-sided rectangles
[article]

1994
*
arXiv
*
pre-print

We show how to determine if

arXiv:math/9411216v1
fatcat:5mbpczf62fgk5cvmonxolhtreq
*a*given simple rectilinear polygon can be*tiled**with*rectangles, each having an integer side. ... Suppose we want to*tile**a**simply**connected*polyomino*with*rectangles, each*of*which has*a*side*of**length*n. ... In the case*of*polyominoes, this lattice is just Z, and so the algorithm yields in this case*a**tiling**with**bars**of**length*1 × n and n × 1. ...##
###
Page 6220 of Mathematical Reviews Vol. , Issue 99i
[page]

1999
*
Mathematical Reviews
*

The main results

*of*the paper are the following two algorithms:*a*linear al- gorithm*of**tiling**a**simply**connected**figure**with*“leaning*bars*”*of**length*m (i.e. parallelograms*with*large sides*of**length*... m and small sides*of*unit*length*) and equilateral triangles*with*sides*of**length*m as well as*a*quadratic algorithm*of**tiling**with*leaning*bars*. ...##
###
Tiling with bars under tomographic constraints
[article]

2001
*
arXiv
*
pre-print

We wish to

arXiv:cs/9903020v3
fatcat:hbq52uzdbvbptnofgxpq4dggdq
*tile**a*rectangle*or**a*torus*with*only vertical and horizontal*bars**of**a*given*length*, such that the number*of**bars*in every column and row equals given numbers. ... We present results for particular instances and for*a*more general problem, while leaving open the initial problem. ... In the same way, each V j is tillable by using only vertical*bars*R 1×v*with*each column having m*bars*.*Figure**3*:*A*(*2*· · ·*2*,*3*· · ·*3*)-*tiling**of*T 15×10 by R*2*×1 and R 1×*3*. ...##
###
A Note on Tiling with Integer-Sided Rectangles

1996
*
Journal of combinatorial theory. Series A
*

We show how to determine if

doi:10.1006/jcta.1996.0053
fatcat:ygfj4c4q2jfhfpe7c2juiras2q
*a*given simple rectilinear polygon can be*tiled**with*rectangles, each having an integer side. ... Suppose we want to*tile**a**simply**connected*polyomino*with*rectangles, each*of*which has*a*side*of**length*n. ...*Figure*1 : 1 Inductive step in the proof that*a**tiling*gives*a*product*of*lassos.*Figure**2*:*2*Removing an interior maximum.*Figure**3*:*3*Lowest and highest*tilings**of**a*polygon. ...##
###
Phase Transitions in Random Dyadic Tilings and Rectangular Dissections

2018
*
SIAM Journal on Discrete Mathematics
*

We consider

doi:10.1137/17m1157118
fatcat:trdh4oa6vren3j4c6xzgcxlicq
*a*weighted version*of*these Markov chains where, given*a*parameter λ > 0, we would like to generate each rectangular dissection (*or*dyadic*tiling*) σ*with*probability proportional to λ |σ| , ...*A*similar edge-flipping chain is also known to*connect*the state space when restricted to dyadic*tilings*, where each rectangle is required to have the form where s, t, u and v are nonnegative integers. ... The*lengths**of*the vertical boundary segments*of*G are integer multiples*of*h, and G is*simply**connected*as it is the union*of**a**connected*region*with*all*of*its holes. ...##
###
Phase Transitions in Random Dyadic Tilings and Rectangular Dissections
[chapter]

2014
*
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
*

We consider

doi:10.1137/1.9781611973730.104
dblp:conf/soda/CannonMR15
fatcat:f2v4kgjh4vhgjnmtrhjqv3neba
*a*weighted version*of*these Markov chains where, given*a*parameter λ > 0, we would like to generate each rectangular dissection (*or*dyadic*tiling*) σ*with*probability proportional to λ |σ| , ...*A*similar edge-flipping chain is also known to*connect*the state space when restricted to dyadic*tilings*, where each rectangle is required to have the form where s, t, u and v are nonnegative integers. ... The*lengths**of*the vertical boundary segments*of*G are integer multiples*of*h, and G is*simply**connected*as it is the union*of**a**connected*region*with*all*of*its holes. ...##
###
Tiling Rectangles with Gaps by Ribbon Right Trominoes

2017
*
Open Journal of Discrete Mathematics
*

We show that the least number

doi:10.4236/ojdm.2017.72010
fatcat:i5dr3oocdnhkhbvcmpowvlvgbe
*of*cells (the gap number) one needs to take out from*a*rectangle*with*integer sides*of**length*at least*2*in order to be*tiled*by ribbon right trominoes is less than*or*equal ... If the sides*of*the rectangle are*of**length*at least 5, then the gap number is less than*or*equal to*3*. ... For any*simply**connected*region that can be*tiled*by ∑*or*∑ , the difference between the number*of*1 T*tiles*and the number*of**2*T*tiles*that appear in the*tiling*is an invariant. ...##
###
Tiling simply connected regions with rectangles

2013
*
Journal of combinatorial theory. Series A
*

Although #P-completeness is known for

doi:10.1016/j.jcta.2013.06.008
fatcat:3ihbnp43drgojopcevxojzfzgi
*tilings**of*general regions*with*right tromino and square tetromino [MR], nothing was known for*tilings**with*rectangles,*or*for*tilings**of**simply**connected*regions. ... Theorem 1.2 There exists*a*finite set R*of*at most 10 6 rectangular*tiles*, such that counting the number*of**tilings**of**simply**connected*regions*with*R is #P-complete. ... We make*a*stronger conjecture that for every tileset T*of*two*bars*[1 × k] and [ℓ × 1], where k, ℓ ≥*2*, (k, ℓ) = (*2*,*2*), the counting*of**tilings*by T*of**simply**connected*regions is #P-complete. ...##
###
Fixed Parameter Undecidability for Wang Tilesets

2012
*
Electronic Proceedings in Theoretical Computer Science
*

Deciding if

doi:10.4204/eptcs.90.6
fatcat:itwhfqjuzvhaze5gegwu2orxs4
*a*given set*of*Wang*tiles*admits*a**tiling**of*the plane is decidable if the number*of*Wang*tiles*(*or*the number*of*colors) is bounded, for*a*trivial reason, as there are only finitely many such ... One*of*the main new tool is the concept*of*Wang*bars*, which are equivalently inflated Wang*tiles**or*thin polyominoes. ... Acknowledgements The first author thanks Daniel Gonçalves and Pascal Vanier for some interesting discussions that lead to the proof*of*the case*of**2*Wang*Bars*. ...##
###
Tiling figures of the plane with two bars

1995
*
Computational geometry
*

Given two "

doi:10.1016/0925-7721(94)00015-n
fatcat:yw7pubjwlrf5ffiuqp756qdwpa
*bars*",*a*horizontal one, and*a*vertical one (both*of**length*at least two), we are interested in the following decision problem: is*a*finite*figure*drawn on*a*plane grid tilable*with*these*bars*... Given*a*general pair*of**bars*, we give two results: (1)*a*necessary condition to have*a*unique*tiling*for finite*figures*without holes, (*2*)*a*linear algorithm (in the size*of*the*figure*) deciding whether ... There exists*a*polynomial algorithm which: (1) decides the existence*of**a**tiling**of**a*f nite*figure*F*with*$*2*,*or*not. (*2*) provides*a**tiling**of*F*with*$*2*,/f such*a**tiling*exists. ...##
###
Tiling with Bars and Satisfaction of Boolean Formulas

1996
*
European journal of combinatorics (Print)
*

In this lattice , the problem

doi:10.1006/eujc.1996.0042
fatcat:spxmkp2uzfc6pbji6jwc4qh2my
*of**tiling**a**simply**connected**figure**with**2*-*bars*can be solved in the linear time (see [6] ) , and the problem*of**tiling**a**figure**with**3*-*bars*seems to be very dif ficult , ... The problem*of**tiling**a**figure**with**bars**of**length**2**or**3*can be reduced in linear time to the logic problem*2*-SAT . P ROPOSITION*3*. 1 . Let F be*a*finite*figure*. ...
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