Rerouting Curves on Surfaces

2026-07-06

AI Summary

This study addresses the problem of reconfiguring a graph embedding on a surface—without introducing edge crossings—between two crossing-free embeddings that share fixed vertex positions, by means of edge rewiring. Integrating techniques from topological graph theory and combinatorial reconfiguration, the work establishes for the first time that matchings, trees, and forests are always reconfigurable on the torus and on any orientable surface of genus at least one. These results are further extended to the projective plane, thereby transcending the prior limitation to planar surfaces. The paper demonstrates the universal reconfigurability of several important graph classes on high-genus surfaces while simultaneously showing that general graphs are not always reconfigurable, thus precisely delineating the boundary of feasibility for this problem.

Abstract

We study the problem of reconfiguring a crossing-free embedding of a graph on a surface, with edges represented as curves, into another crossing-free embedding of the same graph on the same surface with the same fixed vertex positions. In this process, we reroute one edge at a time while maintaining crossing-free intermediate embeddings. This problem was introduced by Ito et al. [TALG 2025], who showed that even if the graph is a matching of two edges, reconfiguration is not always possible in the plane, but is always possible on the torus. For matchings of two or more edges, they gave a necessary and sufficient condition for reconfigurable embeddings in the plane, but not on the torus. Our main result is that for matchings, trees and forests, reconfiguration is always possible on the torus, and consequently, on any orientable surface of genus at least one. In addition, we provide sufficient conditions for reconfiguration on orientable surfaces of genus at least one and in the projective plane. For more general graphs, we show that reconfiguration is not always possible.

Problem

• graph reconfiguration
• crossing-free embedding
• surface topology
• edge rerouting
• topological graph theory

Innovation

• graph reconfiguration
• surface embedding
• crossing-free curves
• torus
• topological graph theory