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A polyomino P is called
*L*-convex if for every two cells there exists a monotone path included in P with at most one change of direction. This paper is a theoretical step for the reconstruction of all
*L*-convex polyominoes by using the geometrical paths. First we investigate the geometrical properties of all subclasses of non-directed
*L*-convex polyominoes by giving nine geometries that characterize all non-directed
*L*-convex polyominoes. Finally, we study the subclasses of directed
*L*-convex polyominoes and we give necessary and sufficient conditions for polyominoes to be
*L*-convex.

A planar discrete set is a finite subset of the integer lattice ℕ 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 [

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 (or HV-convex) if it is both row and column-convex (see

A directed polyomino is obtained by starting out from a cell called source and by adding some other cells in two pre-determined directions, for example east and south, that is, to the right of, or below, the existing cells. A directed convex polyomino is a directed polyomino having connected columns and rows (see

In this paper we study the geometrical aspects of a particular family of convex polyominoes, introduced in [

This class of polyominoes has been considered from different points of view. In [

A problem frequently studied in literature is the reconstruction of a discrete set, on which some connectivity constraints are imposed, from partial informations. In particular, Discrete Tomography considers the problem of reconstructing a discrete set from measurements, generically known as projections, of the number of cells in the set that lie on lines with fixed scopes. In the special case of a convex polyomino P, one considers orthogonal (horizontal and vertical) projections, i.e. the pair ( H , V ) that gives the number of cells in each column and row of P, respectively. In [

This paper is divided into 5 sections. After basics on polyominoes, we investigate in Section 3 the geometrical properties between the feet of all subclasses of non-directed L-convex polyominoes by giving nine geometries. Then these geometries are simplified to four by creating the link between all of them. Finally, using these four geometries we give a theorem that allows us to control the L-convexity of all non-directed convex polyominoes. In Section 4, we introduce the properties of all subclasses of directed L-convex polyominoes and we give the conditions of the L-convexity. A final comment on these geometrical properties is given in Section 5.

To each discrete set S, represented as an 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 [

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 ∈ 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 it is entirely made of only two of the four types of steps defined above [

Proposition 1 (Gastiglione, Restivo). [

In this section, we investigate 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 [ min ( S ) , max ( S ) ] ( [ min ( E ) , max ( E ) ] , [ min ( N ) , max ( N ) ] , [ min ( W ) , max ( 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 [ min ( E ) , max ( E ) ] , [ min ( N ) , max ( N ) ] and [ min ( W ) , max ( 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 | min ( N ) = min ( S ) and min ( W ) = min ( 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 | ( min ( N ) < min ( S ) or min ( N ) > min ( S ) ) and ( min ( W ) < min ( E ) or min ( W ) > min ( 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 exists an L-path from min ( N ) to max ( E ) with a south step followed by an east step, and an L-path from min ( W ) to max ( S ) with an east step followed by a south step.

Proof. It is an immediate result from the fact that P is an L-convex and P is not a directed polyomino (see

Proposition 4. Let P be an L-convex polyomino in the class α L , then the feet of P are connected by at least an L-path from min ( N ) to max ( S ) with a south step followed by an east step, and an L-path from min ( W ) to max ( E ) with an east step followed by a south step (see

Proof. It is an immediate result from the fact that P is an L-convex and P is not a directed polyomino (see

Proposition 5. Let P be an L-convex polyomino in the class β L , then at least one of the two following affirmations is true.

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

2) The feet of P are connected by an L-path from min ( N ) to max ( S ) with a south step followed by an east step and an L-path from max ( W ) to min ( E ) with an east step followed by a north step (see

Proof. Here the result comes from the fact that P is an L-convex and P is not a directed polyomino and we consider one of the two cases (1) min ( W ) < min ( E ) or (2) min ( W ) > min ( E ) (see

Proposition 6. Let P be an L-convex polyomino in the class γ L , then at least one of the two following affirmations is true.

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

2) The feet of P are connected by an L-path from min ( W ) to max ( E ) with an east step followed by a south step and an L-path from max ( N ) to min ( S ) with a south step followed by a west step (see

Proof. Here the result comes from the fact that P is an L-convex and P is not a directed polyomino and we consider one of the two cases 1) max ( S ) < max ( N ) or 2) max ( S ) > max ( N ) (see

Proposition 7. 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 min ( N ) to max ( S ) with an east step followed by a south step and an L-path from min ( W ) to max ( E ) with a south step followed by an east step.

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

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

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

Proof. Here the result comes from the fact that P is an L-convex and P is not a directed polyomino and we consider one of the four cases (1) min ( W ) < min ( E ) and max ( S ) < max ( N ) or (2) min ( W ) > min ( E ) and max ( S ) < max ( N ) or (3) min ( W ) < min ( E ) and max ( S ) > max ( N ) or (4) min ( W ) > min ( E ) and max ( S ) > max ( N ) (see

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

Proposition 8. 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 .

Proof. This is a direct summary of the last four propositions (see

Remark 1. The geometries ( 1 ) ∩ ( 4 ) , ( 2 ) ∩ ( 5 ) , ( 2 ) ∩ ( 6 ) , and ( 3 ) ∩ ( 5 ) mentioned in Proposition 8 give directly the two L-paths mentioned in Proposition 3.

The geometries ( 2 ) ∩ ( 4 ) , ( 3 ) ∩ ( 4 ) , and ( 3 ) ∩ ( 6 ) in Proposition 8 give directly the L-path from min ( N ) to max ( E ) with a south step followed by east step.

The geometries ( 1 ) ∩ ( 5 ) and ( 1 ) ∩ ( 6 ) in Proposition 8 give directly the L-path from min ( W ) to max ( S ) with an east step followed by a south step.

Now, we define the cells on the SE and WE 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 I = { ( i 1 , j 1 ) , ( i 2 , j 2 ) , ⋯ , ( i r , j r ) } be the set of cells belonging to P such that ( i 1 , j 1 ) = ( m , max ( S ) ) , ( i r , j r ) = ( max ( E ) , n ) , and for 2 ≤ k ≤ r − 1 , let ( i k , j k ) be the cells situated on the border of the set SE.

Similarly, let J = { ( i ′ 1 , j ′ 1 ) , ( i ′ 2 , j ′ 2 ) , ⋯ , ( i ′ s , j ′ s ) } be the set of cells belonging to P such that such that ( i ′ 1 , j ′ 1 ) = ( m , min ( S ) ) , ( i ′ s , j ′ s ) = ( max ( W ) ,1 ) , and for 2 ≤ l ≤ s − 1 , let ( i ′ l , j ′ l ) be the cells situated on the border of the set WS.

Now let X = { x 1 , ⋯ , x k , ⋯ , x r } be the set of cells such that x 1 = ( m − v max ( S ) + 1 , max ( S ) ) , ⋯ , x k = ( i k − v j k + 1 , j k ) , ⋯ , x r = ( min ( E ) , n ) and Z = { z 1 , ⋯ , z k , ⋯ , z r } be the set of cells such that z 1 = ( m , min ( S ) ) , ⋯ , z k = ( i k , j k − h i k + 1 ) , ⋯ , z r = ( max ( E ) , n − h max ( E ) + 1 ) .

Similarly, let X ′ = { x ′ 1 , ⋯ , x ′ l , ⋯ , x ′ s } be the set of cells such that x ′ 1 = ( m − v min ( S ) + 1 , min ( S ) ) , ⋯ , x ′ l = ( i l − v j l + 1 , j l ) , ⋯ , x ′ s = ( min ( W ) , 1 ) and Z ′ = { z ′ 1 , ⋯ , z ′ l , ⋯ , z ′ s } be the set of cells such that z ′ 1 = ( m , max ( S ) ) , ⋯ , z ′ l = ( i ′ l , j l + h i l − 1 ) , ⋯ , z ′ s = ( max ( W ) , 1 + h max ( W ) − 1 ) (see

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

• ( 2 ) ∩ ( 5 ) ∈ α

• ( 2 ) ∩ ( 4 ) ∈ β

• ( 2 ) ∩ ( 6 ) ∈ β

• ( 1 ) ∩ ( 5 ) ∈ γ

• ( 3 ) ∩ ( 5 ) ∈ γ

• ( 1 ) ∩ ( 4 ) ∈ μ

• ( 1 ) ∩ ( 6 ) ∈ μ

• ( 3 ) ∩ ( 4 ) ∈ μ

• ( 3 ) ∩ ( 6 ) ∈ μ .

Then P is an L-convex polyomino if and only if for 2 ≤ k ≤ r − 1 , 2 ≤ l ≤ s − 1 the cells situated at the positions ( m − v max ( S ) , min ( S ) − 1 ) , ⋯ , ( i k − v j k , j k − h i k ) , ⋯ , ( min ( E ) − 1 , n − h max ( E ) ) and ( m − v min ( S ) , max ( S ) + 1 ) , ⋯ , ( i l − v j l , j l + h i l ) , ⋯ , ( min ( W ) − 1 , 1 + h max ( W ) ) 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 ( m − v max ( S ) , min ( S ) − 1 ) , ⋯ , ( i k − v j k , j k − h i k ) , ⋯ , ( min ( E ) − 1 , n − h max ( E ) ) and ( m − v min ( S ) , max ( S ) + 1 ) , ⋯ , ( i l − v j l , j l + h i l ) , ⋯ , ( min ( W ) − 1 , 1 + h max ( W ) ) 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 ( m − v max ( S ) , min ( S ) − 1 ) , ⋯ , ( i k − v j k , j k − h i k ) , ⋯ , ( min ( E ) − 1 , n − h max ( E ) ) and ( m − v min ( S ) , max ( S ) + 1 ) , ⋯ , ( i l − v j l , j l + h i l ) , ⋯ , ( min ( W ) − 1 , 1 + h max ( W ) ) control maximal rectangles from SE and WS. Thus, they control the L-convexity of the polyomino (see

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.

• δ = { P ∈ C | ( 1 , min ( N ) ) = ( min ( W ) , 1 ) } .

• ψ = { P ∈ C | ( max ( W ) , 1 ) = ( m , min ( S ) ) } .

• δ ′ = { P ∈ C | ( m , max ( S ) ) = ( max ( E ) , n ) } .

• ψ ′ = { P ∈ C | ( 1 , max ( N ) ) = ( min ( E ) , n ) } .

• δ L = { P ∈ C L | ( 1 , min ( N ) ) = ( min ( W ) , 1 ) } (see

• ψ L = { P ∈ C L | ( max ( W ) , 1 ) = ( m , min ( S ) ) } .

• δ ′ L = { P ∈ C L | ( m , max ( S ) ) = ( max ( E ) , n ) } (see

• ψ ′ L = { P ∈ C L | ( 1 , max ( N ) ) = ( min ( E ) , n ) } .

Let us define the horizontal transformation (symmetry)

S H : ( i , j ) → ( m − i + 1, j )

which transforms the polyomino P from δ to ψ , δ ′ to ψ ′ , δ L to ψ L , and δ ′ L to ψ ′ L . Indeed the transformation acts on the feet of the polyomino as it is shown in the following table (see

Proposition 9. Let P be an L-convex polyomino in the class δ L , then there exist two L-paths from min ( N ) = min ( W ) to max ( E ) with a south step followed by an east step, and from min ( N ) = min ( W ) to max ( S ) with an east step followed by a south step.

Proof. The two L-paths control the L-convexity of the feet (see

Theorem 2. Let P be a convex polyomino in the class δ such that there exist two L-paths from min ( N ) = min ( W ) to max ( E ) with a south then an east steps, and from min ( N ) = min ( W ) to max ( S ) with an east then a south steps. Then P is an L-convex polyomino if and only if the cell at the position ( max ( W ) + 1, max ( N ) + 1 ) does not belong to P.

Proof. The maximal rectangle from the point ( 1,1 ) has extremal cells in the position ( max ( W ) , max ( N ) ) . That is the point at position

N, S | W, E |
---|---|

S → N N → S W → W E → E | W → W E → E W → W E → E |

( max ( W ) + 1, max ( N ) + 1 )

is reachable from cell ( 1,1 ) by a 2L-path. Thus the point at the position ( max ( W ) + 1, max ( N ) + 1 ) does not belong to P (see

Proposition 10. Let P be an L-convex polyomino in the class δ ′ L , then there exist two L-paths from max ( E ) = max ( S ) to min ( N ) with a west step followed by a north step, and from max ( E ) = max ( S ) to min ( W ) with a north step 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 max ( E ) = max ( S ) to min ( N ) with a west step followed by a north step, and from max ( E ) = max ( S ) to 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 min ( E ) − 1, min ( S ) − 1 does not belong to P (see

In this paper we studied the geometrical properties of all subclasses of directed and non-directed L-convex polyominoes and we gave necessary and sufficient conditions to characterize them. The results of this paper will be used in order to reconstruct all L-convex polyominoes using geometrical paths. This paper may help us to understand the geometrical behavior of kL-convex polyominoes and hence find a way to reconstruct them all.

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

Tawbe, K. and Mansour, S. (2019) L-Convex Polyominoes: Geometrical Aspects. Applied Mathematics, 10, 646-658. https://doi.org/10.4236/am.2019.108046