For polyominoes coded by their boundary word, we describe a quadratic O(n) algorithm in the boundary length n which improves the naive O(n) algorithm. Techniques used emanate from algorithmics, discrete geometry and combinatorics on words.

Perfect matchings in chemical graphs correspond to Kekulé structures underlying structure, which play an important role the analysis of resonance energy and stability certain compounds. In 2007, Vukičević Trinajstić introduced concept anti-Kekulé number be smallest edges that have removed from a graph such it remains connected but without any perfect matching. Thereafter, this parameter attract...

Last year, a new notion of rep-cube was proposed. A rep-cube is a polyomino that is a net of a cube, and it can be divided into some polyominoes such that each of them can be folded to a cube. This notion was inspired by the notions of polyomino and rep-tile, which were introduced by Solomon W. Golomb. It was proved that there are infinitely many distinct rep-cubes. In this paper, we investigat...

We exhibit a canonical connection between maximal (0, 1)-fillings of a moon polyomino avoiding northeast chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity result for Schubert polynomials. For ...

A polyomino P is called 2-convex if for every two cells there exists a monotone path included in P with at most 2 changes of direction. This paper studies the tomographical aspects of two subclasses of 2-convex polyominoes called η2L and η′ 2L. In the first part, the uniqueness results of the two subclasses of HV convex polyominoes η and η′ are investigated using the switching components (that ...

Skyline queries are important in many application domains. In this paper, we propose a novel structure Skyline Diagram, which given a set of points, partitions the plane into a set of regions, referred to as skyline polyominos. All query points in the same skyline polyomino have the same skyline query results. Similar to kth-order Voronoi diagram commonly used to facilitate k nearest neighbor (...

We describe a class of geometric objects called k-omino towers that are created by stacking blocks of size k×1×1 on a convex base composed of these same k-omino blocks. These towers are equivalent to a class of fixed polyominoes or lattice animals, through the creation of a polyomino using the boundary of the two-dimensional vertical face of the tower. By applying a partition to the set of k-om...

The set of the three dimensional polyominoes of minimal area and of volume n contains a polyomino which is the union of a quasicube j × (j + δ) × (j + θ), δ, θ ∈ {0, 1}, a quasisquare l × (l + ), ∈ {0, 1}, and a bar k. This shape is naturally associated to the unique decomposition of n = j(j +δ)(j +θ)+ l(l+ )+k as the sum of a maximal quasicube, a maximal quasisquare and a bar. For n a quasicub...