Reconstruction of 2-Convex Polyominoes with Non-Empty Corners

This paper uses the theoretical material developed in a previous study by the authors in order to reconstruct a subclass of 2-convex polyominoes called 1,1 2L α where the upper left corner and the lower right corner of the polyomino contain each only one cell. The main idea is to control the shape of these polyominoes by using 32 types of geometries. Some modifications are made in the reconstruction algorithm of Chrobak and Dürr for HV-convex polyominoes in order to impose these geometries.


Introduction
The present paper uses the theoretical material developed in a previous study by the authors in order to reconstruct a subclass of 2-convex polyominoes. Indeed, 2-convex polyominoes are the first difficult class of polyominoes in terms of tomographical reconstruction in the hierarchy of k-polyominoes and in this article we design an algorithm of reconstruction for a subclass of L-convex which is the second step in the whole comprehension of the hierarchy of k-polyominoes.
One main problem in discrete tomography consists on the reconstruction of discrete objects according to their horizontal and vertical projection vectors. In order to restrain the number of solutions, we could add convexity constraints 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 are polyominoes with 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 k-convex if for every two cells we find a monotone path with at most k changes of direction. Obviously a k-convex polyomino is an HV-convex polyomino. Thus, the set of k-convex polyominoes for k ∈  forms a hierarchy of HV-convex polyominoes according to the number of changes of direction of monotone paths. This notion of 1-convex polyominoes has been considered by several points of view.
In [4] combinatorial aspects of 1-convex polyominoes are analyzed, giving the enumeration according to the semi-perimeter and the area. In [5] it is given an algorithm that reconstructs a 1-convex polyomino from the set of its maximal L-polyominoes. Similarly in [3] it is given another way to reconstruct a L-convex polyomino from the size of some special paths, called bordered L-paths.
In fact 2-convex polymoninoes are more geometrically complex and there was no result for their direct reconstruction. We could notice that Duchi, Rinaldi, and Schaeffer are able to enumerate this class in an interesting and technical article [6]. But the enumeration technique gives no idea for the tomographical reconstruction.
The first subclass that creates the link with 2-convex polyominoes is the class of HV-centered polyominoes. In [7], it is showed that if P is an HV-centered polyomino then P is 2-convex. Note that the tomographical properties of this subclass have been studied in [8] and its reconstruction algorithm is well known.
The main contribution of this paper is an ( ) 3 3 O m n -time algorithm for reconstructing a subclass of 2-convex polyominoes using the geometrical properties studied in a previous study, and the algorithm of Chrobak and Dürr [8]. In particular, we add well-chosen clauses to the original construction of Chrobak and Dürr in order to control the 2L-convexity using 2SAT satisfaction problem. This paper is divided into 5 sections. After basics on polyominoes, Section 3 shows the different geometrical shapes of a subclass of 2-convex polyominoes [9].
In Section 4, the algorithm of Chrobak and Dürr for the reconstruction of the HV-convex polyominoes is given [8]. Section 5 describes the reconstruction of different subclasses of 2-convex polyominoes.

Definition and Notation
A planar discrete set is a finite subset of the integer lattice 2  defined up to 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 Figure 1. A finite set of ×   , and its representation in terms of a binary matrix and a set of cells.
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 (or HV-convex) if it is both row and column-convex (see Figure 2).
To each discrete set S, represented as a m n × binary matrix, two integer vectors ( )  are associated such that, for each 1 ,1 i m j n ≤ ≤ ≤ ≤ , 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 3). Moreover if S has H and V as horizontal and vertival projections, respectively, then we say that S satisfies (H, V). Using the usual matrix notations, the element ( , ) i j denotes the entry in row i and column j.
For any two cells A and B in a polyomino, a path AB Π , from A to B, is a sequence ( ) ( ) ( ) Finally, we define a path to be monotone if it is entirely made of only two of the four types of steps defined above (see [7] [10]). Proposition 1 (Castiglione, Restivo). A polyomino P is HV-convex if and only if every pair of cells is connected by a monotone path.
Let us consider a polyomino P. A path in P has a change of direction in the We call k-convex an HV-convex polyomino such that every pair of its cells can be connected by a monotone path with at most k changes of direction respectively.
In [5], it is proposed a hierarchy on convex polyominoes based on the number of changes of direction in the paths connecting any two cells of a polyomino.  path having at most one single change of direction. In the present study, we focus our attention on the next level of the hierarchy, i.e. the class of 2-convex polyominoes, whose geometrical properties are more complicated and harder to be investigated than those of 1-convex polyominoes (see Figure 4).

2-Convex Polyominoes
Let ( ) , H V be two projection vectors and let P be an HV-convex polyomino, that satisfies ( ) , H V . By a classical argument P is contained in a rectangle R (called minimal bounding box) where in this box no projection gives a zero. Let  ) be the intersection of P's boundary on the lower (right, upper, left) side of R (see [1]). By abuse of notation, we call Figure 5).
 are called E-foot, N-foot and W-foot. For a bounding rectangle R and for a given polyomino P, let us define the following sets: Let  be the class of HV-convex polyominoes, thus we have the following classes of polyominoes regarding the position of the non-intersecting feet.  Figure 6).  Figure 6).  Figure 6).  Figure 6).
where P is a 2-convex polyomino.   γ and hence it is in the class 1,1 2L α (see Figure 7).   δ and hence it is in the class 1,1 2L α (see Figure 8).     α (see Figure 9). Theorem 4. Let P be a convex polyomino in the class 1 Corollary 4. If P satisfies the conditions of Theorem 5, then P is in the class

HV-Convex Polyominoes
Assume that H, V denote strictly positive row and column sum vectors. We also assume that The idea of Chrobak and Dürr [8] for reconstructing an HV-convex polyomino is 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 way other corner regions can be defined. Let P be the complement of P. The definition of HV-convex polyominoes directly implies the following lemma.
The set of clauses Dis means that all four corners are pairwise disjoint, that 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 ( ) 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  Finally, by defining an horizontal symmetry H S , we show how to reconstruct P in the class 1,1 2L β . γ in order to reconstruct them.

Clauses for the Class
: for symbols , , , , , In order to reconstruct all 2-convex polyominoes in the class 1        LGEO  LGEO

Clauses for the Class
In order to reconstruct all 2-convex polyominoes in the class 1,1