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A polyomino P is called 2-convex if for every two cells there exists a monotone path included in P with at most two changes of direction. This paper studies the geometrical properties of a sub-class of 2-convex polyominoes called where the upper left corner and the lower right corner of the polyomino each contain s only one cell.

Discrete convexity intervenes in many domains with regard to geometry and particularly to image processing.

Many notions of discrete convexity of polyominoes were investigated (see [

An HV-convex polymino is called k-convex if for every two cells one can find a monotone path with at most k changes of direction. When the value of k is equal to 1, we have the class of 1-convex polyominoes. This notion of 1-convex polyominoes has been investigated by different researchers (see [

In [

From now on, 2-convex polyominoes which are not L-convex and HV-centered are considered.

In [

Another subclass of 2-convex polyominoes has been studied in [

Directed 2-convex polyominoes with empty corners have been studied in [

In this paper, the study of the 2-convex polyominoes subclasses continues by investigating the geometrical properties of another subclass called α 2 L 1,1 where the lower right corner and the upper left corner are non-empty and each contains only one cell.

This paper is divided into 4 sections. In the first section, an introduction on the different works done before on convex, 1-convex, and some sublcasses of 2-convex polyominoes is given. In the second section, different notations on the feet of the polyominoes are introduced in order to understand the geometrical shapes of the class α 2 L 1,1 . In the third section, 32 geometries in the class α 2 L 1,1 are given. The last section is reserved for the future work.

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

A polyomino P is a plane geometric figure formed by joining one or more equal squares edge to edge. 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 HV-convex if it is both row and column-convex (see [

To each discrete set S, represented as a m × n binary matrix, two integer vectors H = ( h 1 , ⋯ , h m ) and V = ( v 1 , ⋯ , v n ) are associated such that, for each 1 ≤ i ≤ m ,1 ≤ j ≤ n , h i and v j are the number of cells of S 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

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 = ( i 1 , j 1 ) and B = ( i r , j r ) in a polyomino P, a path from A to B, is a sequence of adjacent disjoint cells belonging to P. For each 1 ≤ k ≤ r − 1 , 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 .

Definition 1. A path in a polyomino P is said to be monotone if it is entirely made of the four types of steps defined above (see [

Proposition 1 (Castiglione, Restivo [

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 .

Definition 2. An HV-convex polyomino is said to be k-convex if every pair of its cells can be connected by a monotone path with at most k changes of direction respectively.

In the present study, we focus our attention on the class of 2-convex polyominoes, whose geometrical properties are more complicated and harder to be investigated than those of 1-convex polyominoes (see

In this section, we study the geometrical aspects of 2-convex polyominoes in terms of positions of the feet. 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 3. 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 (see [

Definition 4. Let P be an HV-convex polyomino, we say that P is h-centered [resp. v-centered], if its W-foot and E-foot [resp. N-foot and S-foot] intersect, that is there at least one row going from one foot to another (see

The following property links h-centered polyominoes or v-centered polyominoes to 2-convex polyominoes :

Proposition 2. If P is an h-centered polyomino or a v-centered polyomino, then it is a 2-convex polyomino (see [

Let C be the class of convex polyominoes thus we have two classes of polyominoes regarding the position of the non-intersecting feet.

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

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

Let us define the horizontal transformation (symmetry) [

The study of the classes α and β is very complicated until now. So we make the choice of studying one special case called α 1,1 where the lower right corner and the upper left corner of the polyomino contain each only one cell (see

describe the geometries of 2-convex polyominoes in the class α 1,1 and to give characterizations of such 2-convex polyominoes in terms of paths.

For a bounding rectangle R and for a given polyomino P, 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 above sets with the classes α and β allow us to define the following two classes:

• α 1 , 1 = { P ∈ C | max ( N ) < min ( S ) and max ( W ) < min ( E ) ; c a r d ( W N ) = 1 and c a r d ( S E ) = 1 } , (see

• β 1 , 1 = { P ∈ C | max ( S ) < min ( N ) and max ( E ) < min ( W ) ; c a r d ( N E ) = 1 and c a r d ( W S ) = 1 } , (see

• α 2 L 1 , 1 = { P ∈ C | max ( N ) < min ( S ) and max ( W ) < min ( E ) ; c a r d ( W N ) = 1 and c a r d ( S E ) = 1 } , where P is a 2-convex polyomino.

• β 2 L 1 , 1 = { P ∈ C | max ( S ) < min ( N ) and max ( E ) < min ( W ) ; c a r d ( N E ) = 1 and c a r d ( W S ) = 1 } , where P is a 2-convex polyomino.

Note that the horizontal symmetry S H maps α 2 L 1,1 to β 2 L 1,1 .

The following characterization holds for convex polyominoes in the class α 1,1 .

Theorem 1. Let P be a convex polyomino in the class α 1,1 , P is 2-convex if and only if there exist nine paths:

1) from min ( N ) to max ( E ) ,

2) and from min ( N ) to max ( S ) ,

3) and from min ( W ) to max ( E ) ,

4) and from min ( W ) to max ( S ) ,

5) and from the corner cell in WN to max ( E ) ,

6) and from the corner cell in WN to max ( S ) ,

7) and from min ( N ) to the corner cell in SE,

8) and from min ( W ) to the corner cell in SE,

9) and from the corner cell in WN to the corner cell in SE, having at most two changes of direction.

Proof. (Þ) It is an immediate consequence of the definition of 2-convex polyominoes.

(Ü) Suppose that P is not 2-convex, then there exist two cells ( i , j ) and ( i ′ , j ′ ) such that any path between them has more than two changes of direction. Let us suppose that ( i , j ) is at the position ( min ( W ) ≤ i ≤ max ( W ) ,1 ) and ( i ′ , j ′ ) is at the position ( m , min ( S ) ≤ j ′ ≤ max ( S ) ) (the other positions are similar). We have the following cases.

CASE 1:

If the path from min ( W ) to the corner cell in SE has one change of direction, i.e. there exists an L-path between them, then by convexity there is an L-path between ( i , j ) and ( i ′ , j ′ ) , hence the contradiction.

CASE 2:

If the path from min ( W ) to the corner cell in SE has two changes of direction, one can observe the following cases.

• Either the path goes through min ( S ) and then there exist an L-path between min ( W ) and min ( S ) , thus by convexity there exists a 2L-path from ( i , j ) to ( i ′ , j ′ ) , hence the contradiction; or

• The path goes through max ( W ) and then there is an L-path between max ( W ) and the corner cell in SE, thus there exists a 2L-path from ( i , j ) to ( i ′ , j ′ ) , hence the contradiction; or

• The path goes through max ( N ) and then there is an L-path between max ( N ) and the corner cell in SE, thus there exists a 2L-path from ( i , j ) to ( i ′ , j ′ ) , hence the contradiction; or

• The path goes through a path where its vertical position is between max ( N ) and min ( S ) , thus there exists a 2L-path from ( i , j ) to ( i ′ , j ′ ) , hence the contradiction (see

The cases (1), (2), (3), (4), (5), (6), (7) are similar up to symmetry. □

Corollary 1. If P satisfies Theorem 1, then P is in the class α 2 L 1,1 (see

Now in order to understand the different geometries of the class α 2 L 1,1 let us define the following subclasses of the class α 1,1 :

• γ 1 , 1 = { P ∈ C | max ( N ) < min ( S ) and max ( W ) < min ( E ) ; c a r d ( W N ) = 1 and c a r d ( S E ) = 1 ; min ( E ) = max ( w ) + 1 and min ( S ) = max ( N ) + 1 } (see

• δ 1 , 1 = { P ∈ C | max ( S ) < min ( N ) and max ( E ) < min ( W ) ; c a r d ( N E ) = 1 and c a r d ( W S ) = 1 ; min ( E ) ≠ max ( W ) + 1 and min ( S ) = max ( N ) + 1 } . (see

• Δ 1 , 1 = { P ∈ C | max ( N ) < min ( S ) and max ( W ) < min ( E ) ; c a r d ( W N ) = 1 and c a r d ( S E ) = 1 ; min ( E ) = max ( w ) + 1 and min ( S ) ≠ max ( N ) + 1 } (see

• χ 1 , 1 = { P ∈ C | max ( N ) < min ( S ) and max ( W ) < min ( E ) ; c a r d ( W N ) = 1 and c a r d ( S E ) = 1 ; min ( E ) ≠ max ( w ) + 1 and min ( S ) ≠ max ( N ) + 1 } (see

• γ 2 L 1 , 1 = { P ∈ C | max ( N ) < min ( S ) and max ( W ) < min ( E ) ; c a r d ( W N ) = 1 and c a r d ( S E ) = 1 ; min ( E ) = max ( w ) + 1 and min ( S ) = max ( N ) + 1 } , where P is a 2-convex polyomino.

• δ 2 L 1 , 1 = { P ∈ C | max ( S ) < min ( N ) and max ( E ) < min ( W ) ; c a r d ( N E ) = 1 and c a r d ( W S ) = 1 ; min ( E ) ≠ max ( W ) + 1 and min ( S ) = max ( N ) + 1 } , where P is a 2-convex polyomino.

• Δ 2 L 1 , 1 = { P ∈ C | max ( N ) < min ( S ) and max ( W ) < min ( E ) ; c a r d ( W N ) = 1 and c a r d ( S E ) = 1 ; min ( E ) = max ( w ) + 1 and min ( S ) ≠ max ( N ) + 1 } , where P is a 2-convex polyomino.

• χ 2 L 1 , 1 = { P ∈ C | max ( N ) < min ( S ) and max ( W ) < min ( E ) ; c a r d ( W N ) = 1 and c a r d ( S E ) = 1 ; min ( E ) ≠ max ( w ) + 1 and min ( S ) ≠ max ( N ) + 1 } , where P is a 2-convex polyomino.

Theorem 2. Let P be a convex polyomino in the class γ 1,1 , P is 2-convex if and only if there exists an L-path from

1)

{ min ( N ) to min ( E ) and the corner cell in W N to min (S)

or

2)

{ min ( W ) to min ( S ) and the corner cell in W N to min (E)

or

3)

{ max ( N ) to max ( E ) and max ( W ) to the corner cell in S E

or

4)

{ max ( W ) to max ( S ) and max ( N ) to the corner cell in S E

Proof. (Ü) Suppose that P satisfies only the first geometry, i.e. there exist L-paths from min ( N ) to min ( E ) and from the corner cell in WN to min ( S ) . From the first L-path, one can deduce that there exist 2L-paths from min ( N ) to max ( E ) , from min ( N ) to max ( E ) and from min ( N ) to the corner cell in SE. Now from the second L-path, one can deduce that there exist 2L-paths from min ( W ) to max ( S ) , from min ( W ) to max ( E ) , from min ( W ) to the corner cell in SE, from the corner cell in WN to max ( E ) , from the corner cell in WN to max ( S ) , and finally from the corner cell in WN to the corner cell in SE. To summarize, all nine paths in Theorem 1 are in P and hence P is 2-convex.

Similar reasoning holds for the geometries (2), (3), and (4).

(Þ) P is 2L-convex polyomino then, there exist 2L-paths from min ( W ) to max ( E ) , from min ( N ) to max ( S ) , from min ( N ) to the corner cell in SE, from min ( W ) to the corner cell in SE, from the corner cell in WN to max ( E ) , and from the corner cell in WN to max ( S ) . Now, suppose that the four minimal geometries in

1) There exist L-paths from min ( N ) to the corner cell in SE and from max ( W ) to min ( S ) . From the first L-path, one can deduce that there exists an L-path between max ( N ) and the corner cell in SE. From the second L-path, one can see that there is no information between min ( W ) and max ( S ) , hence P is not 2L-convex polyomino.

2) There exist L-paths from the corner cell in WN to max ( S ) and from max ( N ) to min ( E ) . From the first L-path, one can deduce that there is an L-path between the corner cell in WN to min ( S ) . From the second L-path, one can see that there is no information between min ( N ) and max ( E ) , hence P is not 2L-convex polyomino.

3) There exist L-paths from min ( W ) to the corner cell in SE and from max ( N ) to min ( E ) . From the first L-path, one can deduce that there exists an L-path between max ( N ) and the corner cell in SE. From the second L-path, one can see that there is no information between min ( N ) and max ( E ) , hence P is not 2L-convex polyomino.

4) There exist L-paths from the corner cell in WN to max ( E ) and from max ( W ) to min ( S ) . From the first L-path, one can deduce that there is an L-path between the corner cell in WN to min ( S ) . From the second L-path, one can see that there is no information between min ( W ) and max ( E ) , hence P is not 2L-convex polyomino.

In conclusion, the four geometries are necessary to characterize a 2L-convex polyomino in the class γ 1,1 (see

Corollary 2. If P satisfies the conditions of Theorem 2, then P is in the class γ 2 L 1,1 and hence in the class α 2 L 1,1 .

Theorem 3. Let P be a convex polyomino in the class δ 1,1 , P is 2-convex if and only if there exists an L-path from

1)

{ min ( N ) to min ( E ) and the corner cell in W N to min (S)

or

2)

{ max ( W ) to max ( S ) and max ( N ) to the corner cell in S E

or

3)

{ max ( N ) to max ( E ) and max ( W ) to the corner cell in S E

or

4)

{ min ( W ) to min ( S ) and the corner cell in W N to min (E)

or

5)

{ max ( N ) to max ( E ) and the corner cell in W N to min ( S ) and 2 L -path starting by a south step from min ( N ) to the corner cell in S E

or

6)

{ min ( W ) to min ( S ) and max ( N ) to the corner cell in S E and 2 L -path starting by asouth step from the corner cellin W N to max (S)

or

7)

{ min ( W ) to min ( S ) and max ( N ) to max ( E ) and 2 L -path starting by asouth step from the corner cellin W N to the corner cell in S E

Proof. Similar reasoning as in Theorem 2 (see

Corollary 3. If P satisfies the conditions of Theorem 3, then P is in the class δ 2 L 1,1 and hence in the class α 2 L 1,1 .

Theorem 4. Let P be a convex polyomino in the class Δ 1,1 , P is 2-convex if and only if there exists an L-path from

1)

{ min ( W ) to min ( S ) and the corner cell in W N to min (E)

or

2)

{ max ( N ) to max ( E ) and max ( W ) to the corner cell in S E

or

3)

{ max ( W ) to max ( S ) and max ( N ) to the corner cell in S E

or

4)

{ min ( N ) to min ( E ) and the corner cell in W N to min (S)

or

5)

{ max ( W ) to max ( S ) and the corner cell in W N to min ( E ) and 2 L -pathstartingbyaneaststepfrom min ( W ) to the corner cell in S E

or

6)

{ min ( N ) to min ( E ) and max ( W ) to the corner cell in S E and 2 L -path starting by an east step from the corner cellin W N to max (E)

or

7)

{ min ( N ) to min ( E ) and max ( W ) to max ( S ) and 2 L -path starting by an east step from the corner cellin W N to the corner cell inSE

Proof. Similar reasoning as in Theorem 2 (see

Corollary 4. If P satisfies the conditions of Theorem 4, then P is in the class Δ 2 L 1,1 and hence in the class α 2 L 1,1 .

Theorem 5. Let P be a convex polyomino in the class χ 1,1 , P is 2-convex if and only if there exists an L-path from

1)

{ max ( N ) to max ( E ) and max ( W ) to the corner cell in S E

or

2)

{ min ( N ) to min ( E ) and min ( W ) to the corner cell in S E and 2 L -path starting by an east step from the corner cellin W N to max (E)

or

3)

{ min ( N ) to min ( E ) and the corner cell in W N to min (S)

or

4)

{ max ( N ) to max ( E ) and the corner cell in W N to min ( S ) and 2 L -path starting by a south step from min ( N ) to the corner cell in S E

or

5)

{ max ( W ) to max ( S ) and max ( N ) to the corner cell in S E

or

6)

{ min ( W ) to min ( S ) and max ( N ) to the corner cell in S E and 2 L -path starting by asouth step from the corner cellin W N to max (S)

or

7)

{ min ( W ) to min ( S ) and the corner cell in W N to min (E)

or

8)

{ max ( W ) to max ( S ) and the corner cell in W N to min ( E ) and 2 L -pathstartingbyaneaststepfrom min ( W ) to the corner cell in S E

or

9)

{ max ( W ) to max ( S ) and min ( N ) to min ( E ) and 2 L -path starting by an east step from the corner cellin W N to the corner cell in S E

or

10)

{ max ( W ) to max ( S ) and max ( N ) to max ( E ) and 2 L -path starting by an east step from to th ecorner cellin W N to the corner cell in S E and 2 L -path starting by a south step from min ( N ) to the corner cell in S E

or

11)

{ min ( W ) to min ( S ) and min ( N ) to min ( E ) and 2 L -path starting by an east step from to th ecorner cellin W N to the corner cell in S E and 2 L -path starting by asouth step from the corner cellin W N to max (S)

or

12)

{ max ( N ) to max ( E ) and min ( W ) to min ( S ) and 2 L -path starting by asouth step from the corner cellin W N tothecornercell in S E

or

13)

{ max ( W ) to max ( S ) and max ( N ) to max ( E ) and 2 L -path starting by asouth step from the corner cellin W N tothecornercell in S E and 2 L -pathstartingbyaneaststepfrom min ( W ) to the corner cell in S E

or

14)

{ min ( W ) to min ( S ) and min ( N ) to min ( E ) and 2 L -path starting by asouth step from the corner cellin W N to the corner cell in S E and 2 L -path starting by an east step from the corner cellin W N to max (E)

Proof. Similar reasoning as in Theorem 2 (see Figures 17-20). □

Corollary 5. If P satisfies the conditions of Theorem 5, then P is in the class χ 2 L 1,1 and hence in the class α 2 L 1,1 .

This study showed that 32 geometries are necessary to characterize all 2-convex polyominoes in the class α 1,1 where the upper left corner and the lower right corner each contains only one cell. Also this study is a theoretical step for the reconstruction of the sub-class α 2 L 1,1 . In the spirit of discrete tomography, the design of a reconstruction algorithm for such polyominoes would be the subject of a future article.

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

Tawbe, K., Ghandour, N. and Atwi, A. (2019) 2-Convex Polyominoes: Non-Empty Corners. Open Journal of Discrete Mathematics, 9, 33-51. https://doi.org/10.4236/ojdm.2019.92005