Reconstruction of 2-convex polyominoes

There are many notions of discrete convexity of polyominoes (namely hvconvex [1], Q-convex [2], L-convex polyominoes [5]) and each one has been deeply studied. One natural notion of convexity on the discrete plane leads to the definition of the class of hv-convex polyominoes, that is polyominoes with consecutive cells in rows and columns. In [1] and [6], it has been shown how to reconstruct in polynomial time hv-convex polyominoes from their horizontal and vertical projections. In addition to that, hv-convex polyominoes have been characterized by the presence, between each pair of its cells, of an internal path having at most two kinds of unit steps among the four possible ones, i.e. north, south, east and west steps (such a path is called monotone). For each k ∈ N, we consider the hv-convex polyominoes where each couple of cells can be connected by a monotone path having at most k changes of direction, and we call them k-convex polyominoes. Since each hv-convex polyomino P has a finite number of pairs of cells, and so a finite number of monotone paths connecting them, then there exists an integer k such that P is k′-convex, for each k′ ≥ k. Thus, the families of k-convex polyominoes forms a hierarchy of hv-convex polyominoes. When the value of k is equal to 1 we have the so called L-convex polyominoes, where this terminology is motivated by the L-shape of the path having one single change of direction that connects any two of its cells. This notion of L-convex polyominoes has been considered by several points of view. In particular it has been shown that they are characterized by their horizontal and vertical projections [4]. Other tomographical and combinatorial aspects of L-convex polyominoes have been analyzed in [3] and [5].