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Polyominoes defined by two vectors

1994
*
Theoretical Computer Science
*

Now we give the notion of satisfiability to permit the definition of the

doi:10.1016/0304-3975(94)90107-4
fatcat:mryyd6x7p5aotfnluwsfqvor7q
*polyominoes*' problem*defined**by**two**vectors*, as proposed*by*M. Nivat. ... We*define*ccld;f'bI, Let AE N" and BE N" be*two**vectors*. The pair (A, B) is satisfiable in 23V%? ... case where aI<a2 and anfan_l and b,<bz and b,<b,_I, we would have completely solved the*polyominoes*problem*defined**by**two**vectors*in $7. ...##
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Polyominoes defined by their vertical and horizontal projections

1996
*
Theoretical Computer Science
*

Therefore, every subclass of these

doi:10.1016/0304-3975(95)00205-7
fatcat:wdaujxpdyfeudlwwz5ckygypi4
*polyominoes*is*defined**by*their orthogonal projections. We consider the enumeration of these subclasses according to the number of their columns and rows. ... In this paper, we prove that there are not*two*directed column-or row-convex*polyominoes*having the same vertical and horizontal projections (V,Z-Z). ... A*polyomino*is a connected finite set of adjacent cells lying*two**by**two*along a side. A*polyomino*is*defined*up to a translation. ...##
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Six bijections between deco polyominoes and permutations
[article]

2008
*
arXiv
*
pre-print

In this paper we establish six bijections between a particular class of

arXiv:0810.2876v1
fatcat:gof7jhmx6rcrbgpu462dtlt6ki
*polyominoes*, called deco*polyominoes*, enumerated according to their directed height*by*n!, and permutations. ... Each of these bijections allows us to establish different correspondences between classical statistics on deco*polyominoes*and on permutations. ... A parallelogram*polyomino*can be*defined*as an array of unit cells that is bounded*by**two*lattice paths which use the steps (1, 0) and (0, 1), and which intersect only at their origin and extremity. ...##
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On translating one polyomino to tile the plane

1991
*
Discrete & Computational Geometry
*

If such a tiling exists the

doi:10.1007/bf02574705
fatcat:36dxnvrxoreebk27m7rh5zmtpy
*polyomino*is called an exact*polyomino*. Further, every such tiling of the plane*by*translated copies of the*polyomino*is half-periodic. ... Given a*polyomino*, we prove that we can decide whether translated copies of the*polyomino*can tile the plane. Copies that are rotated, for example, are not allowed in the tilings we consider. ... A tiling T of the plane E*by*P is*defined**by*a subset T of Z 2 such that E is the union of the*polyominoes*Pu for u in T, and, for*two*different*vectors*u and v of T, p. and Pv are disjoint or neighboring ...##
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Total Independent Set Numbers on Catacondensed Polyomino Systems

2021
*
DEStech Transactions on Environment Energy and Earth Science
*

In this paper, we introduce a new graph

doi:10.12783/dteees/peees2020/35460
fatcat:dbdzm6cokjhdfld752qsv7etfq
*vector*at a given edge and get some recurrence relations on the total independent set numbers of the path-like*polyomino*systems. ... A catacondensed*polyomino*system is a chain*polyomino*system in which the joining of the centers of its adjacent cells forms a tree. ... We*define*the Merrifield-Simmons*vector*of G at the edge uv , denoted*by*() uv IG as a column*vector*( ) ( ( ), ( ), ), ( ) . () T uv I G i G i G u i G v i G u v Note that 23 ( ) ( ) ...##
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A Bijection for Directed-Convex Polyominoes

2001
*
Discrete Mathematics & Theoretical Computer Science
*

*polyominoes*having a fixed number of rows and columns. ... International audience In this paper we consider

*two*classes of lattice paths on the plane which use \textitnorth, \textiteast, \textitsouth,and \textitwest unitary steps, beginningand ending at (0,0). ... Most of them can be

*defined*

*by*combining

*two*notions: convexity and directed growth. A

*polyomino*is said to be vertically convex when its intersection with any vertical line is convex. ...

##
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Counting Matching Numbers in Catacondensed Polyomino Systems

2021
*
DEStech Transactions on Computer Science and Engineering
*

The

doi:10.12783/dtcse/ccnt2020/35450
fatcat:rfclph5gyjb4xl36p7ll4hxpna
*polyomino*system is a finite 2connected plane graph such that each interior face (or say a cell) is surrounded*by*a regular square of length one. ... The catacondensed*polyomino*system is a chain*polyomino*system and its central line forms a tree. ... The authors thank the support*by*the National Natural Science Foundation of China (Grant Nos. 11551003), and the Qinghai Natural Science Foundation of China (Grant Nos. 2020-ZJ-924). ...##
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Universal Computation with Arbitrary Polyomino Tiles in Non-Cooperative Self-Assembly
[article]

2014
*
arXiv
*
pre-print

We

arXiv:1408.3351v2
fatcat:r2oydo3w7ffljb2elysjvz4rpi
*define*a generalization of the abstract Tile Assembly Model (aTAM), such that a tile system consists of a collection of*polyomino*tiles, the*Polyomino*Tile Assembly Model (polyTAM), and investigate ... To round out our main result, we provide strong evidence that size-1 (i.e. aTAM tiles) and size-2*polyomino*systems are unlikely to be computationally universal*by*showing that such systems are incapable ... We will*define*the set P 0 as the set of all such paths which can possibly grow. ...##
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L-Convex Polyominoes: Geometrical Aspects

2019
*
Applied Mathematics
*

A

doi:10.4236/am.2019.108046
fatcat:hl35w7htkjh35geoi46q3mmyf4
*polyomino*P is called L-convex if for every*two*cells there exists a monotone path included in P with at most one change of direction. ... This paper is a theoretical step for the reconstruction of all L-convex*polyominoes**by*using the geometrical paths. ... Let ( ) , H V be*two**vectors*of projections and let P be a convex*polyomino*, that satisfies ( ) , H V . ...##
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The number of convex polyominoes reconstructible from their orthogonal projections

1996
*
Discrete Mathematics
*

A

doi:10.1016/s0012-365x(96)83007-x
fatcat:hqmyuzv27zd65gw6pe4uhuzwtm
*polyomino*is a connected finite set of adjacent cells lying*two**by**two*along a side. A polyo~o is*defined*up to a elation. A convex*polyomino*has rows and columns connected. ... ., h,) E N" are*two*assigned*vectors*. ...##
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Counting Polyominoes on Twisted Cylinders

2005
*
Discrete Mathematics & Theoretical Computer Science
*

We achieve this

doi:10.46298/dmtcs.3446
fatcat:h3ufnojykvbnld2shaz4pbt6fy
*by*analyzing*polyominoes*on a different surface, a so-called $\textit{twisted cylinder}$*by*the transfer matrix method. ... A bijective representation of the "states" of partial solutions is crucial for allowing a compact representation of the successive iteration*vectors*for the transfer matrix method. ...*Define*the*vector*x (i) of length |S| with components: x (i) s := {partial*polyominoes*with i occupied cells in state s} (1) Then x (n) {{W }} is the number Z W (n) of n-ominoes on the twisted cylinder ...##
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The Reconstruction of Polyominoes from Approximately Orthogonal Projections
[chapter]

2001
*
Lecture Notes in Computer Science
*

We will prove that it is NPcomplete if we reconstruct

doi:10.1007/3-540-45627-9_22
fatcat:2lgvcfp2rjf5pgmm26fypka5im
*polyominoes*, horizontal convex*polyominoes*and vertical convex*polyominoes*. ... The reconstruction of discrete*two*-dimensional pictures from their projection is one of the central problems in the areas of medical diagnostics, computer-aided tomography, pattern recognition, image processing ... In an analogous way we can*define*other corner regions.*By*P we denote the complement of P . ...##
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A note on polyomino chains with extremum general sum-connectivity index
[article]

2018
*
arXiv
*
pre-print

The general sum-connectivity index of a graph G is

arXiv:1803.04657v1
fatcat:mugjjn24mbcytjgckmzlalem5q
*defined*as χ_α(G)= ∑_uv∈ E(G) (d_u + d_v)^α where d_u is degree of the vertex u∈ V(G), α is a real number different from 0 and uv is the edge connecting ... In this note, the problem of characterizing the graphs having extremum χ_α values from a certain collection of*polyomino*chain graphs is solved for α<0. ... A*polyomino*chain is a*polyomino*system in which every square is adjacent to at most*two*other squares. Every*polyomino*chain can be represented*by*a graph known as*polyomino*chain graph. ...##
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The Number of Line-Convex Directed Polyominoes Having the Same Orthogonal Projections
[chapter]

2006
*
Lecture Notes in Computer Science
*

It is proven that diagonally convex directed

doi:10.1007/11907350_7
fatcat:jibjzbz5snallh65sd4opkkej4
*polyominoes*are uniquely determined*by*their orthogonal projections. The proof of this result is algorithmical. ... The number of line-convex directed*polyominoes*with given horizontal and vertical projections is studied. ... Given*two*integers a and b such that they are coprimes and (a, b) = (0, 0) we*define*the kth line of the discrete rectangle R = {1, . . . , m} × {1, . . . , n} parallel to the*vector*(a, b)*by*S (a,b) ...##
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Universal Computation with Arbitrary Polyomino Tiles in Non-Cooperative Self-Assembly
[chapter]

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

We

doi:10.1137/1.9781611973730.12
dblp:conf/soda/FeketeHPRS15
fatcat:phmky3ldyvctzn2g4yrzwhrbti
*define*a generalization of the abstract Tile Assembly Model (aTAM), such that a tile system consists of a collection of*polyomino*tiles, the*Polyomino*Tile Assembly Model (polyTAM), and investigate ... In order to prove the computational powers of polyTAM systems, we also prove a number of geometric properties held*by*all*polyominoes*of size ≥ 3. ... To show that Γ is simulated*by*some monomino system, we consider*two*cases. For*polyomino*P , let P x denote the*polyomino*obtained*by*translating P*by*some*vector*x. ...
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