L-Convex Polyominoes : Discrete Tomographical Aspects

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.


Introduction
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 [1], Q-convex [2], L-convex polyominoes [3]) and each one leads to interesting studies.One natural notion of convexity on the discrete plane is the class of HV-convex polyominoes that is polyominoes with consecutive cells in rows and columns.Following the work of Del Lungo, Nivat, Barcucci, and Pinzani [1] we are able using discrete tomography to reconstruct polyominoes that are HV-convex according to their horizontal and vertical projections.
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 ∈  forms a hierarchy of HV-convex polyominoes according

Definitions and Notation
A planar discrete set is a finite subset of the integer lattice 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 Figure 1).
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 Figure 2).
A convex polyomino containing at least one corner of its minimal bounding box is said to be a directed convex polyomino.(see Figure 3).
To each discrete set S, represented as a m n × binary matrix, we associate two integer vectors ( ) ≤ ≤ , i h and j v 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 Figure 4).By convention, the origin of the matrix (that is the cell with coordinates ( ) 1,1 ) is in the upper left position.
For any two cells A and B in a polyomino, a path AB Π , from A to B, is a sequence ( ) ( ) ( ) , , , , , , r r i j i j i j  of adjacent disjoint cells ∈ P, with Open Journal of Discrete Mathematics Figure 1.A finite set of ×   , and its representation in terms of a binary matrix and a set of cells (The origin of this figure is in [7]).( ) For each 1 k r ≤ ≤ , we say that the two consecutive cells ( ) ( ) form: • an east step if Let us consider a polyomino P. A path in P has a change of direction in the cell ( ) .
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) [5] A polyomino P is convex if and only if every pair of cells is connected by a monotone path.

Geometrical Properties of L-Convex Polyominoes
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 ( )  ) be the intersection of P's boundary on the lower (right, upper, left) side of R (see [1]). ( ) ] (see Figure 5).
Definition 1.The segment  is called the S-foot.Similarly, the segments  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 ( ) , , , then P is an L-convex polyomino.Proof.Let P be a convex polyomino such that ( )

, , , m H
n h h =  (see Figure 6), then the bar allows us to go from the first cell situated at the position ( ) to all other cells with at most one change of direction.Thus every two cells is connected by a monotone path with at most one change of direction and hence P is an L-convex polyomino.(Similar reasoning holds for the other three cases).
Let  (resp.L  ) be the class of convex polyominoes (resp.L-convex polyominoes) and let P be in  (resp.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: | min min and min min or min min

{ }
| min min or min min and min min | min min or min min and min min or min min | min min and min min

{ }
| min min and min min or min min

{ }
| min min or min min and min min | min min or min min and min min or min min (See Figure 7).
Let us define the following sets: The following characterizations hold for convex polyominoes in the class , , µ α β and L γ .Proposition 3. Let P be an L convex polyomino in the class L µ (resp. , L L α β and L γ ), then there exist an L-path from ( ) max E with a south step followed by an east step, and an L-path from ( ) with an east step followe by a south step.
Proposition 4. Let P be an L-convex polyomino in the class L µ , then at least one of the four following affirmations is true.with a south step followed by an east step.
2) The feet of P are connected by an L-path from ( ) max S with an east step followed by a south step and an L-path from ( ) with an east step followed by a north step.
3) The feet of P are connected by an L-path from ( ) max E with a south step followed by an east step and an L-path from ( ) with an east step followed by a north step.
4) The feet of P are connected by an L-path from ( ) min E with an east step followed by a north step and an L-path from ( ) with an east step followed by a north step (see Figure 8).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 Figure 9.
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 Figure 9.
The geometries ( ) ( ) 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 µ (resp. ,α β and γ ) (P is not directed) and let , , , , , , r r I i j i j i j =  be the set of cells belonging to be the cells situated on the border of the set SE.
Similarly, let , and for ′ ′ be the cells situated on the border of the set WS.
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 the cells situated at the positions , max do not belong to P. Proof.Suppose that P is a convex polyomino.The intersections control the geometries and the L-path between feet.
⇒ If P is an L-convex then obviously the cells situated at the positions do not belong to P. Indeed, these cells could be attained only by using a 2L-path from the SE or WS borders.
⇐ The cells situated at the positions , min , max 1 , , , , , min 1,1 control maximal rectangles from SE and WS.Thus they control the L-convexity of the polyomino (see Figure 11).

Simplification of the Nine Geometries of L-Convex Polyominoes
In this subsection, we show that the four geometries mentionned in Proposition So to reconstruct a non-directed L-convex polyomino we use the combinations of the four L-paths (Figure 12).

Directed L-Convex Polyominoes
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) ( ) ( ) and L δ ′ to L ψ ′ .Indeed the transformation acts on the feet of the polyomino as it is shown in the following table (see Figure 14).Thus we only investigate the properties of the classes L δ and L δ ′ .Proposition 6.Let P be an L-convex polyomino in the class L δ , then there exist two L-paths from   min W with a north followed by a west step.
Theorem 3. Let P be a convex polyomino in the class δ ′ such that there exist two L-paths from ( ) ( ) min N with a west step followed by a north step, annd from ( ) ( ) min W with a north step followed by a west step.Then P is an L-convex polyomino if and only if the cell at the position does not belong to P (see Figure 16).

Reconstruction Algorithms
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.

Chrobak and Dürr's Algorithm
Assume that H, V denote strictly positive row and column sum vectors.We also assume that The idea of Chrobak and Dürr [6] for the control of the HV-convexity is in fact to impose convexity on the four corner regions outside of the polyomino.
An object A is called an upper-left corner region if ( ) In an analogous fashion they can define other corner regions.Let P be the complement of P. The definition of HV-convex polyominoes directly implies the following lemma.

Lemma 1. P is an HV-convex polyomino if and only if P A B C D =    , where , , ,
A B C D are disjoint corner regions (upper-left, upper-right, lower-left and lower-right, respectively) such that 1) ( ) in D, and 2) ( ) Given an HV-convex polyomino P and two row indices 1 ,  k l m ≤ ≤ .P is anchored at ( ) F H V is satisfiable iff there is an HV-convex polyomino realization P of ( ) , H V that is anchored at ( ) F H V consists of several sets of clauses, each set expressing a certain property: "Corners" (Cor), "Disjointness" (Dis), "Connectivity" (Con), "Anchors" (Anc), "Lower bound on column sums" (LBC) and "Upper bound on row sums" (UBR).
, 1, , i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j : for symbols , , , , , 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 ( ) , i j belongs to A then the cell ( ) does not belong to D, and similarly if the cell ( ) , i j belongs to B then the cell ( ) The set of clauses Anc means that we fix two cells on the west and east feet of the polyomino P, for , 1, , k l m =  the first one at the position ( ) and the second one at the position ( ) 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  are assumed to have value 1.

Reconstruction of L-Convex Polyominoes
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 , , : for symbols , , , , , 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. ( , , , min max 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).
GEO3 controls the third geometry ( 3 4  ). ( 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: , 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 , : for symbols , , , , , , Proof.We make the following modifications of the original algorithm of i j i j i j i j i j i j i j i j i j i j i j i j i j : for symbols , , , ,

Figure 2 .
Figure 2. A column convex (left) and a convex (right) polyomino (The origin of this figure is in [3]).

Figure 3 .
Figure 3.A directed convex polyomino (The origin of this figure is in [8]).

Figure 4
Figure 4.A polyomino P with

Figure 5 .
Figure 5. Min and max of the four feet in the rectangle R.

Figure 7 . 1 )
Figure 7. (a) an element of the class L α ; (b) an element of the class L β ; (c) an element

Figure 8 .
Figure 8.The four different L-paths between the feet in the class L µ .

Figure 9 .
Figure 9.The three types of L-paths between each two opposite feet.

2 5 
mentioned in Proposition 5 give directly the two L-paths mentioned in Proposition 3

Figure 10 .
Figure 10.Red cells are the cells situated on the border of SE and WS with

4
are sufficient to reconstruct non-directed L-convex polyominoes in the subclasses , L L α β and L γ and so the nine geometries can be simplified to obtain only four geometries.If could be defined by a point on the larger foot between W-foot and S-foot.If the length of E-foot is larger than the length of W-foot, then we use an L-path between ( ) max W and ( ) min E thus we use the second geometry (1 6  ).If the length of E-foot is smaler than the length of W-foot then we use a L-path between ( ) min W and ( ) max E thus we use the first geometry (1 4 give that we use the third geometry ( 3 4  ) or the fourth geometry ( 3 6  ) depending on the relative length of N-foot and S-foot.

Figure 12 .
Figure 12.The four L-paths between the feet.
an east step followed by a south step.Theorem 2. Let P be a convex polyomino in the class δ such that here exist two L-paths from S with an east followed by a south step.Then P is an L-convex polyomino if and only if the cell at the position to P (see Figure15).

Figure 13 .
Figure 13.An element of the class L δ on the left and one of the class L δ ′ on the right.

Figure 15 .
Figure 15.An L-convex polyomino in the class L δ .
idea of Chrobak and Dürr is, given ( ) , H V , to reconstruct a 2SAT expression (a boolean expression in conjunctive normal form with at most two literals in each clause) . The crucial part of this algorithm comes from the constraints on the two sets of clauses LBC and UBR. is satisfiable if and only if ( ) , H V have a realization P that is an HV-convex polyomino anchored at ( ) , k l .Theorem 5 (Chrobak, Durr) Algorithm 1 solves the reconstruction problem for HV-convex polyominoes in time Figure17).

Figure 17 .
Figure 17.Position, anchored and GEO of the feet in the class L δ .
The set of clauses Cor means that the corners are convex, that is for the corner A if the cell ( )  ∧ DOI: 10.4236/ojdm.2018.84009129 Open Journal of Discrete Mathematics