Order Reconfiguration under Width Constraints

Abstract

In this work, we consider the following order reconfiguration problem: Given a graph $G$ together with linear orders $\omega$ and $\omega'$ of the vertices of $G$, can one transform $\omega$ into $\omega'$ by a sequence of swaps of adjacent elements in such a way that, at each time step, the resulting linear order has cutwidth (pathwidth) at most $k$? We show that this problem always has an affirmative answer when the input linear orders $\omega$ and $\omega'$ have cutwidth (pathwidth) of at most $k/2$. This result also holds in a weighted setting. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory.