Reachability of Independent Sets and Vertex Covers Under Extended Reconfiguration Rules

TL;DR

This paper investigates the computational complexity of reconfiguring independent sets and vertex covers under extended rules, revealing new hardness results and complexity classifications across various graph classes and parameters.

Contribution

It extends the understanding of reconfiguration problems by analyzing the complexity under $k$-Token Jumping and Sliding models, including hardness and membership results for fixed and variable parameters.

Findings

ISR remains NP-hard on graphs of maximum degree 3 and planar graphs of degree 4.

VCR under $k$-TJ is in XP when parameterized by $$.

Both ISR and VCR are PSPACE-complete under various models and graph classes.

Abstract

In reconfiguration problems, we are given two feasible solutions to a graph problem and asked whether one can be transformed into the other via a sequence of feasible intermediate solutions under a given reconfiguration rule. While earlier work focused on modifying a single element at a time, recent studies have started examining how different rules impact computational complexity. Motivated by recent progress, we study Independent Set Reconfiguration (ISR) and Vertex Cover Reconfiguration (VCR) under the $k$-Token Jumping ($k$-TJ) and $k$-Token Sliding ($k$-TS) models. In $k$-TJ, up to $k$ vertices may be replaced, while $k$-TS additionally requires a perfect matching between removed and added vertices. It is known that the complexity of ISR crucially depends on $k$, ranging from PSPACE-complete and NP-complete to polynomial-time solvable. In this paper, we further explore the gradient of computational complexity of the problems. We first show that ISR under $k$-TJ with $k = |I| - \mu$ remains NP-hard when $\mu$ is any fixed positive integer and the input graph is restricted to graphs of maximum degree 3 or planar graphs of maximum degree 4, where $|I|$ is the size of feasible solutions. In addition, we prove that the problem belongs to NP not only for $\mu=O(1)$ but also for $\mu = O(\log |I|)$. In contrast, we show that VCR under $k$-TJ is in XP when parameterized by $\mu = |S| - k$, where $|S|$ is the size of feasible solutions. Furthermore, we establish the PSPACE-completeness of ISR and VCR under both $k$-TJ and $k$-TS on several graph classes, for fixed $k$ as well as superconstant $k$ relative to the size of feasible solutions.