Changing Induced Subgraph Isomorphisms Under Extended Reconfiguration Rules

TL;DR

This paper explores the complexity of reconfiguring induced subgraph isomorphisms when multiple elements can change simultaneously, revealing PSPACE-completeness and providing new theoretical insights.

Contribution

It introduces the study of reconfiguration rules allowing multiple simultaneous changes and establishes complexity results for these scenarios.

Findings

Induced Subgraph Isomorphism Reconfiguration remains PSPACE-complete with multiple changes.

Provides meta-theorems for cases where the number of changed elements is close to the pattern size.

Analyzes complexity of the Independent Set Reconfiguration as a special case.

Abstract

In a reconfiguration problem, we are given two feasible solutions of a combinatorial problem and our goal is to determine whether it is possible to reconfigure one into the other, with the steps dictated by specific reconfiguration rules. Traditionally, most studies on reconfiguration problems have focused on rules that allow changing a single element at a time. In contrast, this paper considers scenarios in which $k \ge 2$ elements can be changed simultaneously. We investigate the general reconfiguration problem of isomorphisms. For the Induced Subgraph Isomorphism Reconfiguration problem, we show that the problem remains $\textsf{PSPACE}$-complete even under stringent constraints on the pattern graph when $k$ is constant. We then give two meta-theorems applicable when $k$ is slightly less than the number of vertices in the pattern graph. In addition, we investigate the complexity of the Independent Set Reconfiguration problem, which is a special case of the Induced Subgraph Isomorphism Reconfiguration problem.