The problem of reconstructing a convex polyominoes from its horizontal and vertical projections when the projections are defined as the number of cells of the polyomino in the different lines and columns was studied by Del Lungo and M. Nivat (cf [7]). In this paper we study the reconstruction of any convex polyomino when the orthogonal projections are defined as the contour length of the object...

In this paper we present an algorithm which has as input a convex polyomino P and computes its degree of convexity, defined as the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. The algorithm uses space O(m + n) to represent a polyomino P with n rows andm columns, and has time complexity O(min(m, rk)), where r is th...

If G is a (molecular) graph with n vertices, and di is the degree of its i-th vertex, then the sum-connectivity index of G is the sum of the weights (du +dv)− 1 2 of all edges uv of G. A polyomino system is a finite 2-connected plane graph such that each interior face (or say a cell) is surrounded by a regular square of length one. Formulas for calculating the sum-connectivity index of polyomin...

Let T be a regular tiling of R which has the origin 0 as a vertex, and suppose that φ : R → R is a homeomorphism such that i) φ(0) = 0, ii) the image under φ of each tile of T is a union of tiles of T , and iii) the images under φ of any two tiles of T are equivalent by an orientation-preserving isometry which takes vertices to vertices. It is proved here that there is a subset Λ of the vertice...

In this paper, we examine rigid tilings of the four quadrants in a Cartesian coordinate system by tiling sets consisting of L-shaped polyominoes and notched rectangles. The first tiling sets we consider consist of an L-shaped polyomino and a notched rectangle, appearing from the dissection of an n×n square, and of their symmetries about the first diagonal. In this case, a tiling of a quadrant i...

We consider variations of the original art gallery problem where the domain is a polyomino, a polycube, or a polyhypercube. Anm-polyomino is the connected union of m unit squares called pixels, an m-polycube is the connected union of m unit cubes called voxels, and an m-polyhypercube is the connected union of m unit hypercubes in a d dimensional Euclidean space. In this paper we generalize and ...

Using memoization and various other optimization techniques, the number of dissections of the n × n square into n polyominoes of size n is computed for n ≤ 8. On this task our method outperforms Donald Knuth’s Algorithm X with Dancing Links. The number of jigsaw sudoku puzzle solutions is computed for n ≤ 7. For every jigsaw sudoku puzzle polyomino cover with n ≤ 6 the size of its smallest crit...