Helly Numbers of Polyominoes

TL;DR

This paper investigates the Helly numbers of polyominoes, establishing that only rectangles have Helly number 2, no polyomino has Helly number 3, and polyominoes with other Helly numbers exist.

Contribution

It characterizes the Helly numbers of polyominoes, proving specific values and the existence or non-existence of polyominoes with certain Helly numbers.

Findings

Rectangles are the only polyominoes with Helly number 2.

No polyomino has Helly number 3.

Polyominoes with Helly numbers other than 1 and 3 exist.

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 $k\neq 1,3$.