Minimal-Perimeter Polyominoes: Chains, Roots, and Algorithms

A polyomino is a set of edge-connected squares on the square lattice. We investigate the combinatorial and geometric properties of minimal-perimeter polyominoes. We explore the behavior of minimal-perimeter polyominoes when they are “inflated,” i.e., expanded by all empty cells neighboring them, and show that inflating all minimal-perimeter polyominoes of a given area create the set of all minimal-perimeter polyominoes of some larger area. We characterize the roots of the infinite chains of sets of minimal-perimeter polyominoes which are created by inflating polyominoes of another set of minimal-perimeter polyominoes, and show that inflating any polyomino for a sufficient amount of times results in a minimal-perimeter polyomino. In addition, we devise two efficient algorithms for counting the number of minimal-perimeter polyominoes of a given area, compare the algorithms and analyze their running times, and provide the counts of polyominoes which they produce.