Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reflexive Digraph Reconfiguration by Orientation Strings

David Emmanuel Pazmiño Pullas, Mark Siggers

TL;DR

This work characterizes the reconfiguration landscape for homomorphisms from digraph cycles C to reflexive cycle targets D by encoding maps as orientation strings and using the wind invariant. It reduces general reconfiguration questions to monotone homomorphisms, developing push-up and refine techniques to prune the search to a monotone core. The authors prove a comprehensive wind_components theorem describing the component structure of Hom_w(C,D) in terms of the primitive root sqrt(D), the exponent r, and a wind parameter w, yielding explicit conditions for when the subgraphs are cyclic or contain multiple components. They also provide linear-time and log-space algorithms to compute the required cycle decompositions and to decide reconfigurability for cyclic instances, establishing that Recon(D) for reflexive cycles can be solved efficiently in log-space for cycle instances. These results advance explicit, space-efficient understanding of digraph homomorphism reconfiguration and have potential implications for broader cycle spaces and topological perspectives on these reconfiguration graphs.

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 $Ω(m^2)$. 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.

Reflexive Digraph Reconfiguration by Orientation Strings

TL;DR

This work characterizes the reconfiguration landscape for homomorphisms from digraph cycles C to reflexive cycle targets D by encoding maps as orientation strings and using the wind invariant. It reduces general reconfiguration questions to monotone homomorphisms, developing push-up and refine techniques to prune the search to a monotone core. The authors prove a comprehensive wind_components theorem describing the component structure of Hom_w(C,D) in terms of the primitive root sqrt(D), the exponent r, and a wind parameter w, yielding explicit conditions for when the subgraphs are cyclic or contain multiple components. They also provide linear-time and log-space algorithms to compute the required cycle decompositions and to decide reconfigurability for cyclic instances, establishing that Recon(D) for reflexive cycles can be solved efficiently in log-space for cycle instances. These results advance explicit, space-efficient understanding of digraph homomorphism reconfiguration and have potential implications for broader cycle spaces and topological perspectives on these reconfiguration graphs.

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 , the algorithm, being greedy, takes time . 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.
Paper Structure (5 sections, 10 theorems, 4 equations, 2 figures)

This paper contains 5 sections, 10 theorems, 4 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Tools for reducing to monotone homomorphisms
  3. Proof of the main theorem via monotone homomorphisms
  4. Algorithms
  5. Concluding Remarks

Key Result

Theorem 1.8

Let $C$ and $D = \sqrt{D}^r = (y_1\dots y_s)^r$ be reflexive digraph cycles such that $D$ is non-contractible. Let $R$ be the maximum value such that $\sigma^i(\sqrt{D}^R) \leq^* C$ for some $i$. Except in the exceptional case that $D$ is a symmetric cycle and $C$ is a directed cycle, the subgraph $ and nothing if $R/r < w$. In the exceptional case, (2) and (3) are replaced with: a single cyclic c

Figures (2)

  • Figure 1: Example of a homomorphism of a digraph $15$-cycle $C$ to a reflexive digraph $4$-cycle $D$
  • Figure 2: Homomorphisms of $C = -++--+-$ to $D = -++--$

Theorems & Definitions (33)

  • Example 1.2
  • Definition 1.3
  • Example 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 23 more