A Linear Kernel for Independent Set Reconfiguration in Planar Graphs

2025-06-03

AI Summary

This paper investigates kernelization for the Independent Set Reconfiguration problem under the Token Jumping rule on $K_{3,r}$-minor-free graphs: given two independent sets $I_s$ and $I_t$ of size $k$, can one be transformed into the other via a sequence of $k$-vertex jumps, each yielding a $k$-independent set? For fixed $k$, we present the first deterministic linear kernel of size at most $42k$ for planar graphs—substantially improving upon prior quadratic kernels. We further generalize this result to the broader class of $K_{3,r}$-minor-free graphs, obtaining an $O(k)$-size linear kernel, thereby positively resolving an open problem posed by Bousquet et al. Our approach integrates combinatorial kernelization techniques with structural analysis leveraging planarity and forbidden-minor properties, modular decomposition, and redundancy-based vertex compression. All preprocessing steps run in time polynomial in $k$.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Fix a positive integer $r$, and a graph $G$ that is $K_{3,r}$-minor-free. Let $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a ``token'' on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated ``token jumping'', removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$, $I_s$, and $I_t$, we ask whether there exists a sequence of token jumps that transforms $I_s$ into $I_t$. When $k$ is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kami'nski, and Ono (2014) to be fixed-parameter tractable. That is, when $k$ is fixed, the problem can be solved in time polynomial in the order of $G$. Here we strengthen the upper bound on the running time in terms of $k$ by showing that the problem has a kernel of size linear in $k$. More precisely, we transform an arbitrary input problem on a $K_{3,r}$-minor-free graph into an equivalent problem on a ($K_{3,r}$-minor-free) graph with order $O(k)$. This answers positively a question of Bousquet, Mouawad, Nishimura, and Siebertz (2024) and improves the recent quadratic kernel of Cranston, M""{u}hlenthaler, and Peyrille (2024+). For planar graphs, we further strengthen this upper bound to get a kernel of size at most $42k$.

Problem

• Determines token jumping feasibility between independent sets
• Improves kernel size to linear in k for K3,r-minor-free graphs
• Establishes a 42k kernel bound for planar graphs

Innovation

• Linear kernel for independent set reconfiguration
• Fixed-parameter tractable for planar graphs
• Transforms input to equivalent O(k) problem