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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.

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 [

Among the theoretical contributions of Golomb to the study of these objects are two tiling hierarchies, one for single polyominoes [

for translations, rotations and reflections in the coordinate axis for all tiles in the set.

Golomb showed all valid implications of tiling capabilities among the levels and that most of them (except for exhibiting a strong rep-tile set which does not tile rectangle and a weak rep-tile set that is not a strong rep-tile set) cannot be reversed.

In [

We can also consider the level of Rep-tile. As we are not able to differentiate it from the level of Rectangle, we will neglect it in the future. In [

When we speak about tiling a DEFICIENT-REGION we mean tiling SOME-DEFICIENT-REGION. This is in the spirit of Golomb’s approach. One can also consider the level ALL-DEFICIENT-REGION. The notions of

SOME-DEFICIENT-PLANE and ALL-DEFICIENT-PLANE coincide, as any two deficient planes are alike, but for other deficient levels the notions of SOME and ALL are distinct. This suggests the addition of the prefixes AD (ALL-DEFICIENT) and SD (SOME-DEFICIENT) to all levels and types of regions considered before, and to study the relationships between their tilings capabilities.

In both Golomb’s and our setups, the impact on tiling hierarchies of the addition of the prefix AD is minimal [

level have independent tiling properties. Second we show that almost all implications of tiling capabilities are invalid. Third we discuss some refinements of the tiling hierarchies if we look at deficient regions and restrict our attention to the class of tile sets that tile rectangles. This restriction allows to recover many of the implications of tiling capabilities that we have for non-deficient regions. All tile sets used in this paper are finite. Many of them are derived from tile sets already used by Golomb in [

A first group of results shows that the types of deficient regions appearing inside each level have independent tiling properties.

Theorem 1. 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.

3) For each three types of deficient half strips there exists a set of polyominoes that tile these types but not the fourth.

Proof. 1) The set of tiles in

2) There are two cases to consider: the half-strips are adjacent and the half-strips are opposite. The set of tiles in

3) The tile set in

In all cases, to show impossibility of tiling, look at the coverings of the corners of the half-strips. This gives a contradiction right away, or leads to the appearance of a half-strip pointing in the wrong direction.

Theorem 2. 1) For each type of deficient bent strip there exists a set of polyominoes that tile that type but not the other three.

2) For each two types of deficient bent strips there exists a set of polyominoes that tile these types but not the other two.

3) For each three types of deficient bent strips there exists a set of polyominoes that tile these types but not the fourth.

Proof. 1) The set of tiles in

2) There are two cases to consider: the bent strips are adjacent and the bent strips are opposite. The tile set in

3) The tile set in

In all cases, to show impossibility of tiling, look at the coverings of the corner of the bent-strips. This leads to an immediate contradiction.

Theorem 3. 1) For each type of deficient quadrant there exists a set of polyominoes that tile that type but not the other three.

2) For each two types of deficient quadrants there exists a set of polyominoes that tile these types but not the other two.

3) For each three types of deficient quadrants strips there exists a set of polyominoes that tile these types but not the other.

Proof. 1) The tile set in

2) There are two cases to consider, depending on the quadrants being adjacent

or opposite. The tile set in

3) The tile set in

In all cases, to show impossibility of tiling, look at the coverings of the corner of the quadrants. This leads to an immediate contradiction. The tilings follow from [

Theorem 4. For each type of deficient strip there exists a set of polyominoes that tile that type but not the other.

Proof. The tile set in

Theorem 5. 1) For each type of deficient half-plane there exists a set of polyominoes that tile that type but not the other three.

2) For each two types of deficient half-plane there exists a set of polyominoes that tile these types but not the other two.

3) For each three types of deficient half-plane there exists a set of polyominoes that tile these types but not the other.

Proof. 1) The tile set in

2) There are two cases to consider, depending on the half-planes intersecting or being opposite. The tile set in

3) The tile set in

The tiling follows from [

A second group of results investigates for deficient regions the validity of the implications of tiling capabilities in in

Theorem 6. The following implications are invalid:

1) DR → DHS(*), where *Î{R, L, U, D};

2) DHS(*) → DQ(**), where *Î{R, L, U, D}, **Î{I, II, III, IV};

3) DQ(*) → DHP(**), where *Î{R, L, U, D}, **Î{I, II, III, IV};

4) DHP(**) → DP, where **Î{I, II, III, IV};.

Proof. 1) A counterexample is given by a tile set consisting of a single deficient rectangle. 2) The tile set in

3) The tile set in

4) The tile set in

We observe that all tile sets used for counterexamples tile not just the respective

deficient regions, but also instances of the respective regions which have an arbitrary finite number of missing cells.

We show now that almost all the implications in

Theorem 7. The following implications are invalid

1) DR → DBS(*), where *Î{I,II,II,IV};

2) DBS(*) → DQ(**), where *,**Î{I,II,II,IV};

3) DS(V) and DQ(*) → DS(H), where *Î{I,II,II,IV};

4) DS(H) and DQ(*) → DS(V), where *Î{I,II,II,IV};

5) DQ(*) → DHP(***), where *Î{I,II,II,IV}, ***Î{U,D,L,R};

6) DHP(***) → DP, where ***Î{U,D,L,R}.

The following implication is valid:

7) DS(H) → DHP(***), where ***Î{U,D};

8) DS(V) → DHP(***), where ***Î{L,R}.

Proof. 1) A counterexample is given by a tile set consisting of a single deficient rectangle.

2) One can use the tile set in

3), 4)

5) This is Theorem 2.1, 3).

6) This is Theorem 2.1, 4).

7), 8) One can show, using the same argument as in the proof of [Theorem 3, 3], that a tile set that tiles a deficient strip of type DS(H) also tiles a horizontal (non-deficient) strip of same width. Then the strips can be repeated on the top or at the bottom of the deficient strip to fill a deficient half plane of type HP(U) or HP(D). A similar argument is valid for 8).

We show in [_{4} consisting of four ribbon L-tetrominoes, with only translations allowed, tile the opposite deficient quadrants DQ(II) and DQ(IV), allowing for any missing cell, but does not tile any of the other two deficient quadrants DQ(I) and DQ(III), no matter what cell is missing. We also show that T_{4} does not tile any deficient half-strip, deficient bent-strip or deficient strip, while it tiles all deficient half-planes and deficient planes. This raises the question if refinements of the tiling hierarchies are possible.

As the tile set T_{4} already tiles rectangles, we study implications of tiling capabilities for deficient regions of tile sets that tile rectangles and consist of simply connected tiles. It is shown in [

Proposition 8. There exists a tile set with all symmetries allowed that tiles some rectangles, some deficient rectangles, and consists of simply connected tiles, but does not tile any other deficient region in Golomb’s hierarchy, except a deficient plane. If only translations are allowed, the tile set does not tile any of the deficient regions in

Proof. The tile set is shown in

Next theorem shows the validity of the rest of the implications in

Theorem 9. The following implications are valid and neither one of them can be reversed:

1) R-and-DHS → R-and-DQ;

2) R-and-DBS → R-and-DQ;

3) R-and-DS → R-and-DHP;

4) R-and-DQ → R-and-DHP;

5) R-and-DHP → R-and-DP.

The following implications are invalid:

6) R-and-DHS → R-and-DBS;

7) R-and-DBS → R-and-DS.

Proof. 1)-5). The proofs for the valid implications are immediate. To show that 1) cannot be reversed use the tile set in

6) The tile set in

7) The tile set in

Theorem 9 allows to recover some of the hierarchy in

Next theorem shows the validity of the rest of the implications in

Theorem 10. If R1 → R2 is an implication in

Proof. The proofs for the valid implications are immediate. To show that the implications cannot be reversed use the tile sets used in Theorem 2.9. Theorem \ref{thm:ref2} allows to recover the hierarchy in

Next theorem shows the validity of the implications in

Theorem 11. The following implications are valid:

1) R-and-DBS(*) → R-and-DQ(*), *Î{I,II,III,IV};

The following implications are invalid:

2) R-and-DBS(*) → R-and-DS(**), *Î{ I,II,III,IV}, **Î{H,V}.

Proof. 1) The proof is immediate.

2) See

The results about tiling deficient regions by T_{4} are based on a coloring invariant introduced in [_{n} can be found in [

V. Nitica was partially supported by Simons Foundation Grant 208729.

Nitica, V. (2018) Revisiting a Tiling Hierarchy (II). Open Journal of Discrete Mathematics, 8, 48-63. https://doi.org/10.4236/ojdm.2018.82005