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Polyominoes defined by two vectors

A. Del Lungo
1994 Theoretical Computer Science  
Now we give the notion of satisfiability to permit the definition of the 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.  ... 
doi:10.1016/0304-3975(94)90107-4 fatcat:mryyd6x7p5aotfnluwsfqvor7q

Polyominoes defined by their vertical and horizontal projections

Elena Barcucci, Alberto Del Lungo, Renzo Pinzani, Renzo Sprugnoli
1996 Theoretical Computer Science  
Therefore, every subclass of these 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.  ... 
doi:10.1016/0304-3975(95)00205-7 fatcat:wdaujxpdyfeudlwwz5ckygypi4

Six bijections between deco polyominoes and permutations [article]

Emeric Deutsch, Elisa Pergola
2008 arXiv   pre-print
In this paper we establish six bijections between a particular class of 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.  ... 
arXiv:0810.2876v1 fatcat:gof7jhmx6rcrbgpu462dtlt6ki

On translating one polyomino to tile the plane

D. Beauquier, M. Nivat
1991 Discrete & Computational Geometry  
If such a tiling exists the 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  ... 
doi:10.1007/bf02574705 fatcat:36dxnvrxoreebk27m7rh5zmtpy

Total Independent Set Numbers on Catacondensed Polyomino Systems

Haizhen Ren, Dongxia Zhu, Deqing Xu
2021 DEStech Transactions on Environment Energy and Earth Science  
In this paper, we introduce a new graph 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 ( ) ( )  ... 
doi:10.12783/dteees/peees2020/35460 fatcat:dbdzm6cokjhdfld752qsv7etfq

A Bijection for Directed-Convex Polyominoes

Alberto Del Lungo, Massimo Mirolli, Renzo Pinzani, Simone Rinaldi
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.  ... 
doi:10.46298/dmtcs.2298 fatcat:uvbpixb6qzb5ncgsysuoouyleq

Counting Matching Numbers in Catacondensed Polyomino Systems

Haizhen Ren, Deqing Xu, Dongxia Zhu
2021 DEStech Transactions on Computer Science and Engineering  
The 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).  ... 
doi:10.12783/dtcse/ccnt2020/35450 fatcat:rfclph5gyjb4xl36p7ll4hxpna

Universal Computation with Arbitrary Polyomino Tiles in Non-Cooperative Self-Assembly [article]

Sándor P. Fekete, Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers, Robert T. Schweller
2014 arXiv   pre-print
We 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.  ... 
arXiv:1408.3351v2 fatcat:r2oydo3w7ffljb2elysjvz4rpi

L-Convex Polyominoes: Geometrical Aspects

Khalil Tawbe, S. Mansour
2019 Applied Mathematics  
A 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 .  ... 
doi:10.4236/am.2019.108046 fatcat:hl35w7htkjh35geoi46q3mmyf4

The number of convex polyominoes reconstructible from their orthogonal projections

Alberto Del Lungo, Maurice Nivat, Renzo Pinzani
1996 Discrete Mathematics  
A 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.  ... 
doi:10.1016/s0012-365x(96)83007-x fatcat:hqmyuzv27zd65gw6pe4uhuzwtm

Counting Polyominoes on Twisted Cylinders

Gill Barequet, Micha Moffie, Ares Ribó, Günter Rote
2005 Discrete Mathematics & Theoretical Computer Science  
We achieve this 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  ... 
doi:10.46298/dmtcs.3446 fatcat:h3ufnojykvbnld2shaz4pbt6fy

The Reconstruction of Polyominoes from Approximately Orthogonal Projections [chapter]

Maciej Gebala
2001 Lecture Notes in Computer Science  
We will prove that it is NPcomplete if we reconstruct 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 .  ... 
doi:10.1007/3-540-45627-9_22 fatcat:2lgvcfp2rjf5pgmm26fypka5im

A note on polyomino chains with extremum general sum-connectivity index [article]

Akbar Ali, Tahir Idrees
2018 arXiv   pre-print
The general sum-connectivity index of a graph G is 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.  ... 
arXiv:1803.04657v1 fatcat:mugjjn24mbcytjgckmzlalem5q

The Number of Line-Convex Directed Polyominoes Having the Same Orthogonal Projections [chapter]

Péter Balázs
2006 Lecture Notes in Computer Science  
It is proven that diagonally convex directed 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)  ... 
doi:10.1007/11907350_7 fatcat:jibjzbz5snallh65sd4opkkej4

Universal Computation with Arbitrary Polyomino Tiles in Non-Cooperative Self-Assembly [chapter]

Sándor P. Fekete, Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers, Robert T. Schweller
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
We 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.  ... 
doi:10.1137/1.9781611973730.12 dblp:conf/soda/FeketeHPRS15 fatcat:phmky3ldyvctzn2g4yrzwhrbti
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