Minimum-Turn Tours of Even Polyominoes

TL;DR

The paper addresses the problem of characterizing minimum-turn tours in even polyominoes $P$ (regions formed by doubling a polyomino). It develops a turn-graph framework, proves that minimum-turn pseudotours are turn-even via a downward phase that deletes cycles to reduce turns, and then uses an upward phase to stitch cycles into a single tour while controlling turn increases. The main result shows that, for $P$ that are topological disks, all minimum-turn tours are regular, confirming Peng and Rascoussier's conjecture in this setting. The approach provides a constructive, two-phase strategy that may inform extensions to regions with holes and contributes to understanding optimal routing structures on grids.

Abstract

Let $P$ be a connected bounded region in the plane formed out of $2 \times 2$ blocks joined by their sides. Peng and Rascoussier conjectured that all minimum-turn Hamiltonian cycles of $P$ exhibit a certain regular structure. We prove this conjecture in the special case when $P$ is a topological disk. The proof proceeds in two phases - a "downward" phase where we break apart an irregular Hamiltonian cycle into a collection of shorter cycles; and an "upward" phase where we put it back together in a different way so that, overall, the number of turns in it decreases.

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