围长为5的图上的记号滑动
Token sliding on graphs of girth five
摘要：
在令牌滑动问题中，我们给出了一个图$G$和两个独立的集$I_s$和$I_t$在$G$的大小$k \geq 1$。目标是决定是否存在独立集合的序列$\langle I_1, I_2, \ldots, I_\ell \rangle$，使得对于所有$i \in \{1,\ldots, \ell\}$，集合$I_i$是大小为$k$、$I_1 = I_s$、$I_\ell = I_t$和$I_i \triangle I_{i + 1} = \{u, v\} \in E(G)$的独立集合。直观地说，我们将每个独立的集合视为放置在图的顶点上的标记的集合。然后，该问题询问是否存在将$I_s$变换为$I_t$的独立集合的序列，其中在每一步中，允许我们将一个令牌从顶点滑动到相邻顶点。在本文中，我们专注于参数化的令牌滑动$k$参数化的复杂性。如Bartier等人所示，这个问题在围长小于等于4的图上是W[1]-困难的，作者提出了是否存在一个常数$p \geq 5$使得这个问题在围长至少为$p$的图上成为固定参数可处理的问题。我们肯定地回答了他们的问题，并证明了这个问题确实是固定参数易处理的围长为5或更大的图，这建立了一个完整的分类令牌滑动的易处理性参数化的令牌数的基础上输入图的围长。
英文摘要：
In the Token Sliding problem we are given a graph $G$ and two independent sets $I_s$ and $I_t$ in $G$ of size $k \geq 1$. The goal is to decide whether there exists a sequence $\langle I_1, I_2, \ldots, I_\ell \rangle$ of independent sets such that for all $i \in \{1,\ldots, \ell\}$ the set $I_i$ is an independent set of size $k$, $I_1 = I_s$, $I_\ell = I_t$ and $I_i \triangle I_{i + 1} = \{u, v\} \in E(G)$. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the problem asks whether there exists a sequence of independent sets that transforms $I_s$ into $I_t$ where at each step we are allowed to slide one token from a vertex to a neighboring vertex. In this paper, we focus on the parameterized complexity of Token Sliding parameterized by $k$. As shown by Bartier et al., the problem is W[1]-hard on graphs of girth four or less, and the authors posed the question of whether there exists a constant $p \geq 5$ such that the problem becomes fixed-parameter tractable on graphs of girth at least $p$. We answer their question positively and prove that the problem is indeed fixed-parameter tractable on graphs of girth five or more, which establishes a full classification of the tractability of Token Sliding parameterized by the number of tokens based on the girth of the input graph.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
