On Girth and the Parameterized Complexity of Token Sliding and Token Jumping

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