Revisiting a Tiling Hierarchy (II)

Viorel Nitica

doi:10.4236/ojdm.2018.82005

Abstract

In a recent paper, we revisited Golomb’s hierarchy for tiling capabilities of finite sets of polyominoes. We considered the case when only translations are allowed for the tiles. In this classification, for several levels in Golomb’s hierarchy, more types appear. We showed that there is no general relationship among tiling capabilities for types corresponding to same level. Then we found the relationships from Golomb’s hierarchy that remain valid in this setup and found those that fail. As a consequence we discovered two alternative tiling hierarchies. The goal of this note is to study the validity of all implications in these new tiling hierarchies if one replaces the simply connected regions by deficient ones. We show that almost all of them fail. If one refines the hierarchy for tile sets that tile rectangles and for deficient regions then most of the implications of tiling capabilities can be recovered.

Highlights

A well-known class of combinatorial objects is that of polyominoes
1) For each type of deficient half strip there exists a set of polyominoes that tile that type but not the other three
2) For each two types of deficient half strips there exists a set of polyominoes that tile these types but not the other two

Summary

Introduction

A well-known class of combinatorial objects is that of polyominoes. A 1 × 1 square in a square lattice is called cell. A polyomino is a connected plane figure built out of a finite number of cells joined along some of their edges. Introduced by Golomb more than half a century ago [1] (but see the monograph [2]), polyominoes continue to generate an endless list of mathematical problems. Among the theoretical contributions of Golomb to the study of these objects are two tiling hierarchies, one for single polyominoes [3] and one for finite sets of polyominoes [4]. The sets of polyominoes classified by Golomb allow

Nitica

Main Results

Conclusion