A binary planar configuration A associates to each point in Z2 an element in {0,1}. Provided a finite window probe P, we locally inspect A by moving P in all its possible positions and counting the 1s elements that fit inside it. In case all the computed values have the same value k, then we say that A is k-homogeneous w.r.t. P. A recent conjecture states that a binary planar configuration is k-homogeneous with respect to an exact polyomino P, i.e., a polyomino that tiles the plane by translation, if and only if it can be decomposed into k configurations that are 1-homogeneous with respect to P. In this paper we define a class of exact polyominoes called perfect pseudo-squares (PPS) and we investigate the periodicity behaviors of the homogeneous configurations that are related to them. Then, we show that some elements in PPS allow 2-homogeneous or 3-homogeneous non-decomposable planar configurations, so providing evidence that the conjecture does not hold for the whole class of exact polyominoes.

On the Decomposability of Homogeneous Binary Planar Configurations with Respect to a Given Exact Polyomino / Ascolese, Michela; Frosini, Andrea. - ELETTRONICO. - 13493:(2022), pp. 139-152. ((Intervento presentato al convegno Discrete Geometry and Mathematical Morphology [10.1007/978-3-031-19897-7_12].

On the Decomposability of Homogeneous Binary Planar Configurations with Respect to a Given Exact Polyomino

Ascolese, Michela;Frosini, Andrea
2022

Abstract

A binary planar configuration A associates to each point in Z2 an element in {0,1}. Provided a finite window probe P, we locally inspect A by moving P in all its possible positions and counting the 1s elements that fit inside it. In case all the computed values have the same value k, then we say that A is k-homogeneous w.r.t. P. A recent conjecture states that a binary planar configuration is k-homogeneous with respect to an exact polyomino P, i.e., a polyomino that tiles the plane by translation, if and only if it can be decomposed into k configurations that are 1-homogeneous with respect to P. In this paper we define a class of exact polyominoes called perfect pseudo-squares (PPS) and we investigate the periodicity behaviors of the homogeneous configurations that are related to them. Then, we show that some elements in PPS allow 2-homogeneous or 3-homogeneous non-decomposable planar configurations, so providing evidence that the conjecture does not hold for the whole class of exact polyominoes.
Discrete Geometry and Mathematical Morphology. DGMM 2022
Discrete Geometry and Mathematical Morphology
Ascolese, Michela; Frosini, Andrea
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2158/1287876
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