Reflexive Digraph Reconfiguration by Orientation Strings

TL;DR

This paper presents a linear-time, log-space algorithm for reconfiguring homomorphisms between cycles to a reflexive digraph cycle, improving efficiency and understanding of the reconfiguration graph's structure.

Contribution

It provides a new characterization of reconfiguration graph components and a linear-time, log-space solution for cycle-to-cycle homomorphism reconfiguration.

Findings

Reconfiguration graph components can be characterized efficiently.

The reconfiguration problem for cycles can be solved in log-space.

A linear-time algorithm for cycle homomorphism reconfiguration is developed.

Abstract

The reconfiguration problem for homomorphisms of digraphs to a reflexive digraph cycle, which amounts to deciding if a `reconfiguration graph' is connected, is known to by polynomially time solvable via a greedy algorithm based on certain topological requirements. Even in the case that the instance digraph is a cycle of length $m$, the algorithm, being greedy, takes time $\Omega(m^2)$. Encoding homomorphisms between two cycles as a relation on strings that represent the orientations of the cycles, we give a characterization of the components of the reconfiguration graph that can be computed in linear time and logarithmic space. In particular, this solves the reconfiguration problem for homomorphisms of cycles to cycles in log-space.