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arXiv · 2510.24226

Reachability of Independent Sets and Vertex Covers Under Extended Reconfiguration Rules

Shuichi Hirahara·Naoto Ohsaka·Tatsuhiro Suga·Akira Suzuki·Yuma Tamura·Xiao Zhou
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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 $k = |I| - μ$ remains NP-hard when $μ$ 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 $μ=O(1)$ but also for $μ= O(\log |I|)$. In contrast, we show that VCR under $k$-TJ is in XP when parameterized by $μ= |S| - k$, 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.

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Shuichi Hirahara, Naoto Ohsaka, Tatsuhiro Suga, Akira Suzuki, Yuma Tamura, Xiao Zhou. 2025-10-28. Reachability of Independent Sets and Vertex Covers Under Extended Reconfiguration Rules. https://arxiv.org/abs/2510.24226

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