发表机构
University of Liverpool; Philipps-Universität Marburg（利物浦大学; 马尔堡菲利普斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究阈值符号图上独立集重配置的令牌跳跃和滑动令牌问题，利用其包含链结构，证明这两个问题在该图类上可在多项式时间内解决，此前这两个问题在一般图上是PSPACE完全的。
AI 中文摘要
令牌跳跃和滑动令牌问题是在无向图的独立集上定义的基本重配置问题。给定两个大小为k的独立集I和J，这些问题询问是否存在一系列基本操作将I转换为J，使得每个中间配置也是大小为k的独立集。在滑动令牌中，操作将令牌从顶点u∈I移动到相邻顶点v∉I；在令牌跳跃中，令牌可以移动到任何顶点v∉I。虽然这两个问题在一般图上都是PSPACE完全的，但已经为几个图类开发了多项式时间算法。本文证明这两个问题在阈值符号图（也称为Dilworth-2图）上都可以在多项式时间内解决。算法基于表征阈值符号图的包含链结构。
英文摘要
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 $k$. Suppose a token is placed on each vertex of $I$: 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 $\mathsf{PSPACE}$-complete on general graphs, polynomial-time algorithms for one or both variants have been developed for several graph classes, including trees, block graphs, 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 $S,T>0$ such that $|a(v)|< \min\{S,T\}$ for all $v \in V$, and for any distinct vertices $u,v\in V$, $\{u,v\}\in E$ if and only if $|a(u)+a(v)|\ge S$ or $|a(u)-a(v)|\ge T$. More generally, we also show that Token Jumping can be solved in time $n^{O(\mathcal{D}(G))}$, where $\mathcal{D}(G)$ denotes the Dilworth number of $G$. Thus, Token Jumping belongs to $\mathsf{XP}$ when parameterised by the Dilworth number. 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).
CommentsOverhauled proofs and structure, and additional XP result
