Helly Numbers of Polyominoes

Abstract

We define the Helly number of a polyomino $P$ as the smallest number $h$ such that the $h$-Helly property holds for the family of symmetric and translated copies of $P$ on the integer grid. We prove the following: (i) the only polyominoes with Helly number 2 are the rectangles, (ii) there does not exist any polyomino with Helly number 3, (iii) there exist polyominoes of Helly number $k$ for any $k\ne 1,3$.

This paper was published in Graphs and Combinatorics, September 2013, Volume 29, Issue 5, pp 1221-1234