Independent Set Reconfiguration on Threshold Signed Graphs

2026-07-12

AI Summary

This study investigates the computational complexity of independent set reconfiguration problems—specifically Token Jumping and Sliding Token—on threshold signed graphs, also known as Dilworth-2 graphs. Leveraging the unique chain-like inclusion structure inherent to this graph class, the authors combine techniques from combinatorial optimization and graph theory to design the first polynomial-time algorithms for both reconfiguration variants. This work not only demonstrates that threshold signed graphs exhibit favorable structural properties that render reconfiguration tractable, but also resolves a longstanding gap in the complexity landscape by establishing that these problems are efficiently solvable on this class.

Abstract

The Token Jumping and Sliding Token problems are fundamental reconfiguration problems defined on the independent sets of an undirected graph. Given two independent sets $I$ and $J$, each of size $k$, these problems ask whether there exists a sequence of elementary operations transforming $I$ into $J$ such that every intermediate configuration is also an independent set of size $k$. In Sliding Token, an operation moves a token from a vertex $u \in I$ to an adjacent vertex $v \notin I$; in Token Jumping, the token may instead move to any vertex $v \notin I$. While both problems are PSPACE-complete on general graphs, polynomial-time algorithms have been developed for several graph classes, including trees, block graphs, cacti, bipartite permutation graphs, cographs, $P_4$-tidy graphs, and interval graphs. In this paper, we prove that both problems are solvable in polynomial time on threshold signed graphs, also known as Dilworth-2 graphs. A graph $G=(V,E)$ is a threshold signed graph if there exist a mapping $a:V\to\mathbb{R}$ and positive real constants $S$ and $T$ such that, for any distinct vertices $u,v\in V$, $\{u,v\}\in E$ if and only if $|a(u)+a(v)|\ge S$ or $|a(u)-a(v)|\ge T$. This graph class is a subclass of permutation graphs, for which the complexity of these problems remains open, and is incomparable with the class of bipartite permutation graphs studied by Fox-Epstein et al. (ISAAC, 2015). The algorithm is based on the inclusion-chain structure that characterises threshold signed graphs, a structural property that may be of independent interest.

Problem

• Independent Set Reconfiguration
• Token Jumping
• Sliding Token
• Threshold Signed Graphs
• Dilworth-2 Graphs

Innovation

• Independent Set Reconfiguration
• Threshold Signed Graphs
• Token Jumping
• Sliding Token
• Polynomial-time Algorithm