Reconfiguring Multiple Connected Components with Size Multiset Constraints

TL;DR

This paper introduces a generalized graph reconfiguration problem called CCR, explores new reconfiguration rules, and provides efficient algorithms for special graph classes, extending prior ISR results.

Contribution

It generalizes ISR to CCR with size multiset constraints, introduces new reconfiguration rules, and offers efficient solutions for specific graph classes.

Findings

CCR is PSPACE-complete in general.

CCR-CS is solvable in linear time on paths.

CCR-CJ is solvable in quadratic time on paths.

Abstract

We propose a novel generalization of Independent Set Reconfiguration (ISR): Connected Components Reconfiguration (CCR). In CCR, we are given a graph $G$, two vertex subsets $A$ and $B$, and a multiset $\mathcal{M}$ of positive integers. The question is whether $A$ and $B$ are reconfigurable under a certain rule, while ensuring that each vertex subset induces connected components whose sizes match the multiset $\mathcal{M}$. ISR is a special case of CCR where $\mathcal{M}$ only contains 1. We also propose new reconfiguration rules: component jumping (CJ) and component sliding (CS), which regard connected components as tokens. Since CCR generalizes ISR, the problem is PSPACE-complete. In contrast, we show three positive results: First, CCR-CS and CCR-CJ are solvable in linear and quadratic time, respectively, when $G$ is a path. Second, we show that CCR-CS is solvable in linear time for cographs. Third, when $\mathcal{M}$ contains only the same elements (i.e., all connected components have the same size), we show that CCR-CJ is solvable in linear time if $G$ is chordal. The second and third results generalize known results for ISR and exhibit an interesting difference between the reconfiguration rules.