TITLE:
Revisiting a Tiling Hierarchy (II)
AUTHORS:
Viorel Nitica
KEYWORDS:
Tiling, Polyomino, Deficient Region, Tile Set, Tiling Hierarchy
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.8 No.2,
April
25,
2018
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.