Abstract

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. First we investigate the geometrical properties of all subclasses of non-directed L-convex polyominoes by giving nine geometries that characterize all non-directed L-convex polyominoes. Finally, we study the subclasses of directed L-convex polyominoes and we give necessary and sufficient conditions for polyominoes to be L-convex.

Highlights

  • A planar discrete set is a finite subset of the integer lattice 2 defined up to translation

  • 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

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Summary

Introduction

A planar discrete set is a finite subset of the integer lattice 2 defined up to translation. In [2] the authors observed that L-convex polyominoes have the property that every pair of cells is connected by a monotone path involving at most one direction. In the special case of a convex polyomino P, one considers orthogonal (horizontal and vertical) projections, i.e. the pair ( H ,V ) that gives the number of cells in each column and row of P, respectively. These geometries are simplified to four by creating the link between all of them.

Definitions and Notations
Geometrical Properties L-Convex Polyominoes
Directed L-Convex Polyominoes
Conclusion
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