In the Token Jumping problem we are given a graph \(G = (V,E)\) and two independent sets S and T of G, each of size \(k \ge 1\). The goal is to determine whether there exists a sequence of k-sized independent sets in G, \(\langle S_0, S_1, \ldots , S_\ell \rangle\), such that for every i, \(|S_i| = k\), \(S_i\) is an independent set, \(S = S_0\), \(S_\ell = T\), and \(|S_i \varDelta S_{i+1}| = 2\). In other words, if we view each independent set as a collection of tokens placed on a subset of the vertices of G, then the problem asks for a sequence of independent sets which transforms S to T by individual token jumps which maintain the independence of the sets. This problem is known to be PSPACE-complete on very restricted graph classes, e.g., planar bounded degree graphs and graphs of bounded bandwidth. A closely related problem is the Token Sliding problem, where instead of allowing a token to jump to any vertex of the graph we instead require that a token slides along an edge of the graph. Token Sliding is also known to be PSPACE-complete on the aforementioned graph classes. We investigate the parameterized complexity of both problems on several graph classes, focusing on the effect of excluding certain cycles from the input graph. In particular, we show that both Token Sliding and Token Jumping are fixed-parameter tractable on \(C_4\)-free bipartite graphs when parameterized by k. For Token Jumping, we in fact show that the problem admits a polynomial kernel on \(\{C_3,C_4\}\)-free graphs. In the case of Token Sliding, we also show that the problem admits a polynomial kernel on bipartite graphs of bounded degree. We believe both of these results to be of independent interest. We complement these positive results by showing that, for any constant \(p \ge 4\), both problems are W[1]-hard on \(\{C_4, \dots , C_p\}\)-free graphs and Token Sliding remains W[1]-hard even on bipartite graphs.

在Token跳跃问题，我们给出的曲线图\（G =（V，E）\）和两组独立的小号和Ť的ģ，各尺寸的\（K \ GE 1 \） 。目标是确定G 中是否存在k大小的独立集序列，\(\langle S_0, S_1, \ldots , S_\ell \rangle\)，使得对于每个i，\(|S_i| = k\) , \(S_i\)是一个独立的集合，\(S = S_0\)，\(S_\ell = T\)和\(|S_i \varDelta S_{i+1}| = 2\). 换句话说，如果我们将每个独立集视为放置在G的顶点子集上的标记的集合，则问题要求将S转换为T的独立集序列通过保持集合独立性的单个令牌跳转。已知这个问题在非常受限的图类上是 PSPACE 完全的，例如，平面有界度图和有界带宽图。一个密切相关的问题是令牌滑动问题，在该问题中，我们要求令牌沿着图的边缘滑动，而不是允许令牌跳转到图的任何顶点。众所周知，令牌滑动在上述图类上是 PSPACE 完备的。我们在几个图类上研究了这两个问题的参数化复杂性，重点关注从输入图中排除某些循环的影响。特别是，我们证明了当由k参数化时，令牌滑动和令牌跳跃在\(C_4\) -free 二部图上都是固定参数可处理的. 对于 Token Jumping，我们实际上表明该问题在\(\{C_3,C_4\}\) 无图上承认多项式核。在令牌滑动的情况下，我们还表明该问题在有界度的二部图上承认多项式核。我们相信这两个结果都具有独立的利益。我们通过表明，对于任何常数\(p \ge 4\)，这两个问题在\(\{C_4, \dots , C_p\}\)上都是 W[1]-hard -free graphs and Token即使在二部图上滑动仍然是 W[1]-hard。