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This paper uses the geometrical properties of L -convex polyominoes in order to reconstruct these polyominoes. The main idea is to modify some clauses to the original construction of Chrobak and Dürr in order to control the L -convexity using 2SAT satisfaction problem.

Discrete tomography focuses on the problem of reconstruction of discrete objects from small number of their projections. In order to reduce the number of solutions we could add some convexity conditions to these discrete objects. There are many notions of discrete convexity of polyominoes (namely HV-convex [

In addition to that, for an HV-convex polyomino P every pairs of cells of P can be reached using a path included in P with only two kinds of unit steps (such a path is called monotone). A polyomino is called kL-convex if for every two cells we find a monotone path with at most k changes of direction. Obviously a kL-convex polyomino is an HV-convex polyomino. Thus, the set of kL-convex polyominoes for k ∈ ℕ forms a hierarchy of HV-convex polyominoes according to the number of changes of direction of monotone paths. This notion of L-convex polyominoes has been considered by several points of view. In [

The main contribution of this paper is the developement of an algorithm that reconstructs all subclasses of L-convex polyominoes by using their geometrical properties and the algorithm of Chrobak and Dürr [

This paper is divided into 6 sections. After basics on polyominoes, I present briefly in Section 3 the four geometrical properties between the feet of all subclasses of non-directed L-convex polyominoes. In Section 4, I also introduce the subclasses of directed L-convex polyominoes with the conditions of the L-convexity. In the last Section I give the reconstruction algorithms of all L-convex polyominoes using simple modifications of Chrobak and Dürr’s algorithm. The last section is a final comment on my contribution.

A planar discrete set is a finite subset of the integer lattice ℤ 2 defined up to a translation. A discrete set can be represented either by a set of cells, i.e. unitary squares of the cartesian plane, or by a binary matrix , where the 1’s determine the cells of the set (see

A polyomino P is a finite connected set of adjacent cells, defined up to translations, in the cartesian plane. A row convex polyomino (resp. column-convex) is a self avoiding convex polyomino such that the intersection of any horizontal line (resp. vertical line) with the polyomino has at most two connected components. Finally, a polyomino is said to be convex if it is both row and column-convex (see

A convex polyomino containing at least one corner of its minimal bounding box is said to be a directed convex polyomino. (see

To each discrete set S, represented as a m × n binary matrix, we associate two integer vectors H = ( h 1 , ⋯ , h m ) and V = ( v 1 , ⋯ , v n ) such that, for each 1 ≤ i ≤ m ,1 ≤ j ≤ n , h i and v j are the number of cells of S (elements 1 of the matrix) which lie on row i and column j, respectively. The vectors H and V are called the horizontal and vertical projections of S, respectively (see

For any two cells A and B in a polyomino, a path Π A B , from A to B, is a sequence ( i 1 , j 1 ) , ( i 2 , j 2 ) , ⋯ , ( i r , j r ) of adjacent disjoint cells Î P, with

A = ( i 1 , j 1 ) , and B = ( i r , j r ) . For each 1 ≤ k ≤ r , we say that the two consecutive cells ( i k , j k ) , ( i k + 1 , j k + 1 ) form:

・ an east step if i k + 1 = i k and j k + 1 = j k + 1 ;

・ a north step if i k + 1 = i k − 1 and j k + 1 = j k ;

・ a west step if i k + 1 = i k and j k + 1 = j k − 1 ;

・ a south step if i k + 1 = i k + 1 and j k + 1 = j k .

Let us consider a polyomino P. A path in P has a change of direction in the cell ( i k , j k ) , for 2 ≤ k ≤ r − 1 , if

i k ≠ i k − 1 ⇔ j k + 1 ≠ j k .

Finally, we define a path to be monotone if its entirely made of only two of the four types of steps defined above.

Proposition 1 (Gastiglione, Restivo) [

In this section, we present the geometrical properties of L-convex polyominoes in terms of monotone paths.

Let ( H , V ) be two vectors of projections and let P be a convex polyomino, that satisfies ( H , V ) . By a classical argument P is contained in a rectangle R of size m × n (called minimal bounding box). Let [ m i n ( S ) , m a x ( S ) ] ( [ m i n ( E ) , m a x ( E ) ] , [ m i n ( N ) , m a x ( N ) ] , [ m i n ( W ) , m a x ( W ) ] ) be the intersection of P’s boundary on the lower (right, upper, left) side of R (see [

Definition 1. The segment [ min ( S ) , max ( S ) ] is called the S-foot. Similarly, the segments [ m i n ( E ) , m a x ( E ) ] , [ m i n ( N ) , m a x ( N ) ] and [ m i n ( W ) , m a x ( W ) ] are called E-foot, N-foot and W-foot.

Proposition 2. Let ( H , V ) be two vectors of projections and let P be a convex polyomino, that satisfies ( H , V ) . If H = ( n , h 2 , ⋯ , h m ) or H = ( h 1 , h 2 , ⋯ , n ) or V = ( m , v 2 , ⋯ , v n ) or V = ( v 1 , v 2 , ⋯ , m ) then P is an L-convex polyomino.

Proof. Let P be a convex polyomino such that H = ( n , h 2 , ⋯ , h m ) (see

Let C (resp. C L ) be the class of convex polyominoes (resp. L-convex polyominoes) and let P be in C (resp. C L ) such that P does not satisfy Proposition 2. Also suppose that P is not a directed polyomino, then one can define the following subclasses of convex polyominoes:

α = { P ∈ C | min ( N ) = min ( S ) and min ( W ) = min ( E ) } .

β = { P ∈ C | min ( N ) = min ( S ) and ( min ( W ) < min ( E ) or min ( W ) > min ( E ) ) } .

γ = { P ∈ C | ( min ( N ) < min ( S ) or min ( N ) > min ( S ) ) and min ( W ) = min ( E ) } .

μ = { P ∈ C | ( min ( N ) < min ( S ) or min ( N ) > min ( S ) ) and ( min ( W ) < min ( E ) or min ( W ) > min ( E ) ) } .

α L = { P ∈ C L | m i n ( N ) = m i n ( S ) and m i n ( W ) = m i n ( E ) } .

β L = { P ∈ C L | min ( N ) = min ( S ) and ( min ( W ) < min ( E ) or min ( W ) > min ( E ) ) } .

γ L = { P ∈ C L | ( min ( N ) < min ( S ) or min ( N ) > min ( S ) ) and min ( W ) = min ( E ) } .

μ L = { P ∈ C L | ( m i n ( N ) < m i n ( S ) or m i n ( N ) > m i n ( S ) ) and ( m i n ( W ) < m i n ( E ) or m i n ( W ) > m i n ( E ) ) } . (See

Let us define the following sets:

・ W N = { ( i , j ) ∈ P ∕ i < min ( W ) and j < min ( N ) } ,

・ S E = { ( i , j ) ∈ P ∕ i > max ( E ) and j > max ( S ) } .

・ N E = { ( i , j ) ∈ P ∕ i < min ( E ) and j > max ( N ) } ,

・ W S = { ( i , j ) ∈ P ∕ i > max ( W ) and j < min ( S ) } .

The following characterizations hold for convex polyominoes in the class μ L , α L , β L and γ L .

Proposition 3. Let P be an L convex polyomino in the class μ L (resp. α L , β L and γ L ), then there exist an L-path from m i n ( N ) to m a x ( E ) with a south step followed by an east step, and an L-path from m i n ( W ) to m a x ( S ) with an east step followe by a south step.

Proposition 4. Let P be an L-convex polyomino in the class μ L , then at least one of the four following affirmations is true.

1) The feet of P are connected by an L-path from m i n ( N ) to m a x ( S ) with an east step followed by a south step and an L-path from m i n ( W ) to m a x ( E ) with a south step followed by an east step.

2) The feet of P are connected by an L-path from m i n ( N ) to m a x ( S ) with an east step followed by a south step and an L-path from m a x ( W ) to m i n ( E ) with an east step followed by a north step.

3) The feet of P are connected by an L-path from m i n ( W ) to m a x ( E ) with a south step followed by an east step and an L-path from m i n ( S ) to m a x ( N ) with an east step followed by a north step.

4) The feet of P are connected by an L-path from m a x ( W ) to m i n ( E ) with an east step followed by a north step and an L-path from m i n ( S ) to m a x ( N ) with an east step followed by a north step (see

Now if P is an L-convex polyomino (P is not directed), then the feet of P are characterized by the geometries shown in the

Case (1) is the first geometry (GEO1 in the algorithm).

Case (2) is the second geometry (GEO2 in the algorithm).

Case (3) is the third geometry (GEO3 in the algorithm).

Case (4) is the fourth geometry (GEO4 in the algorithm).

Proposition 5. Let P be an L-convex polyomino (P is not directed), then the feet of P are connected at least by one of the nine following geometries of the L-paths in

・ ( 2 ) ∩ ( 5 ) ∈ α L

・ ( 2 ) ∩ ( 4 ) ∈ β L

・ ( 2 ) ∩ ( 6 ) ∈ β L

・ ( 1 ) ∩ ( 5 ) ∈ γ L

・ ( 3 ) ∩ ( 5 ) ∈ γ L

・ ( 1 ) ∩ ( 4 ) ∈ μ L

・ ( 1 ) ∩ ( 6 ) ∈ μ L

・ ( 3 ) ∩ ( 4 ) ∈ μ L

・ ( 3 ) ∩ ( 6 ) ∈ μ L .

Remark 1. The geometries ( 1 ) ∩ ( 4 ) , ( 2 ) ∩ ( 5 ) , ( 2 ) ∩ ( 6 ) , and ( 2 ) ∩ ( 5 ) mentioned in Proposition 5 give directly the two L-paths mentioned in Proposition 3.

The geometries ( 2 ) ∩ ( 4 ) , ( 3 ) ∩ ( 4 ) , and ( 3 ) ∩ ( 6 ) in Proposition 5 give directly the L-path from m i n ( N ) to max ( E ) with a south step followed by an east step.

The geometries ( 1 ) ∩ ( 5 ) and ( 1 ) ∩ ( 6 ) in Proposition 5 give directly the L-path from m i n ( W ) to max ( S ) with an east step followed by a south step.

Now, we define the cells on the SE and WS borders to define the sets X , Z , X ′ and Z ′ from these cells.

Let P be a convex polyomino in the class

Similarly, let

Now let

and

Similarly, let

and

(see

Theorem 1. Let P be a convex polyomino such that P satisfies at least one of the following geometries・

・

・

・

Then P is an L-convex polyomino if and only if for

and

do not belong to P.

Proof. Suppose that P is a convex polyomino. The intersections control the geometries and the L-path between feet.

Þ If P is an L-convex then obviously the cells situated at the positions

and

do not belong to P. Indeed, these cells could be attained only by using a 2L-path from the SE or WS borders.

Ü The cells situated at the positions

and

control maximal rectangles from SE and WS. Thus they control the L-convexity of the polyomino (see

In this subsection, we show that the four geometries mentionned in Proposition 4 are sufficient to reconstruct non-directed L-convex polyominoes in the subclasses

If

If

So to reconstruct a non-directed L-convex polyomino we use the combinations of the four L-paths (

Let P be a convex polyomino such that P does not satisfy Proposition 2. From the definition of directed convex polyominoes, let us define the following classes.

・

・

・

・

・

・

・

・

Let us define the horizontal transformation (symmetry)

which transforms the polyomino P from

Proposition 6. Let P be an L-convex polyomino in the class

Theorem 2. Let P be a convex polyomino in the class

Proposition 7. Let P be an L-convex polyomino in the class

Theorem 3. Let P be a convex polyomino in the class

One main problem in discrete tomography consists on the reconstruction of discrete objects according to their vectors of projections. In order to restrain the number of solutions, we could add convexity constraints to these discrete objects. The present section uses the theoretical material presented in the above sections in order to reconstruct all subclasses of L-convex polyominoes. Some modifications are made in the reconstruction algorithm of Chrobak and Dürr for HV-convex

polyominoes in order to impose our geometries. All the clauses that have been added and the modifications of the original algorithm are well explained in the proofs of each subclass.

Assume that H, V denote strictly positive row and column sum vectors. We also assume that

The idea of Chrobak and Dürr [

An object A is called an upper-left corner region if

Lemma 1. P is an HV-convex polyomino if and only if

Given an HV-convex polyomino P and two row indices

The set of clauses Cor means that the corners are convex, that is for the corner A if the cell

The set of clauses Dis means that all four corners are pairwise disjoint, that is

The set of clauses Con means that if the cell

The set of clauses Anc means that we fix two cells on the west and east feet of the polyomino P, for

The set of clauses LBC implies that for each column j, we have that

The set of clauses UBR implies that for each row i, we have that

Define

Algorithm 1.

Input:

W.l.o.g assume:

For

If

then output

end

output “failure”.

The following theorem allows to link the existence of HV-convex solution and the evaluation of

Theorem 4 (Chrobak, Durr)

Theorem 5 (Chrobak, Durr) Algorithm 1 solves the reconstruction problem for HV-convex polyominoes in time

In this subsection, we add the clauses Anc1, COND1, COND2, GEO1, GEO2, GEO3, GEO4, For1 and we modify the clause Anc of the original Chrobak and Dürr’s algorithm in order to reconstruct if it is possible all polyominoes in the subclass

Anc1 is added in order to consider non-directed convex polyominoes by positioning exterior cells of the polyomino in the four corners of the minimal bounding box.

COND1 controls the L-path between E-foot and N-foot (see proposition 3).

COND2 controls the L-path between W-foot and S-foot (see proposition 3).

GEO1 controls the first geometry (

GEO2 controls the second geometry (

GEO3 controls the third geometry (

GEO4 controls the fourth geometry (

For1 controls the cells in the SE and WS borders of P and imposes that the cells of Theorem 1 are outside the polyomino P. In order to reconstruct and to obtain all L-convex polyominoes, we use the set of clauses:

Algorithm 2.

Input:

W.l.o.g assume:

For

If

then output

end

output “failure”.

Proof. The feet of all L-convex polyominoes that are not directed are characterized by at least one of the four geometries described in Theorem 1 and by the property that the cells situated at the positions

do not belong to these polyominoes. Thus we combine all geometries and conditions using suitable set of clauses in order to reconstruct L-convex polyominoes. We make the following modifications of the original algorithm of Chrobak and Dürr [

The set COND1 (resp. COND2) implies that we put a cell in the interior of the polyomino at the position

The set GEO1 implies that we put a cell in the interior of the polyomino at the position

The set GEO2 implies that we put a cell in the interior of the polyomino at the position

The set GEO3 implies that we put a cell in the interior of the polyomino at the position

The set GEO4 implies that we put a cell in the interior of the polyomino at the position

The set For1 implies that the cell

Using the conjunction of the whole set of clauses, if one of the

In this subsection, we add the clauses Pos, GEO, For2 and we modify the clause Anc of the original Chrobak and Dürr’s algorithm in order to reconstruct if it is possible all polyominoes in the class

In order to reconstruct all L-convex polyominoes in the class

Algorithm 3.

Input:

W.l.o.g assume:

For

If

then output

end

output “failure”.

Proof. We make the following modifications of the original algorithm of Chrobak and Dürr in order to add the constraints and the properties of the class

In order to reconstruct all L-convex polyominoes in the class

Algorithm 4

Input:

W.l.o.g assume:

For

If

then output

end output “failure”.

The contribution of this paper will be used to investigate the geometrical and tomographical aspects of kL-convex polyominoes for

The authors declare no conflicts of interest regarding the publication of this paper.

Tawbe, K. and Mansour, S. (2018) L-Convex Polyominoes: Discrete Tomographical Aspects. Open Journal of Discrete Mathematics, 8, 116-136. https://doi.org/10.4236/ojdm.2018.84009