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

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.