Assume we are given a graph $G$, two independent sets $S$ and $T$ in $G$ of size $k \geq 1$, and a positive integer $\ell \geq 1$. The goal is to decide whether there exists a sequence $\langle I_0, I_1, ..., I_\ell \rangle$ of independent sets such that for all $j \in \{0,\ldots,\ell-1\}$ the set $I_j$ is an independent set of size $k$, $I_0 = S$, $I_\ell = T$, and $I_{j+1}$ is obtained from $I_j$ by a predetermined reconfiguration rule. We consider two reconfiguration rules. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the Token Sliding Optimization (TSO) problem asks whether there exists a sequence of at most $\ell$ steps that transforms $S$ into $T$, where at each step we are allowed to slide one token from a vertex to an unoccupied neighboring vertex. In the Token Jumping Optimization (TJO) problem, at each step, we are allowed to jump one token from a vertex to any other unoccupied vertex of the graph. Both TSO and TJO are known to be fixed-parameter tractable when parameterized by $\ell$ on nowhere dense classes of graphs. In this work, we show that both problems are fixed-parameter tractable for parameter $k + \ell + d$ on $d$-degenerate graphs as well as for parameter $|M| + \ell + \Delta$ on graphs having a modulator $M$ whose deletion leaves a graph of maximum degree $\Delta$. We complement these result by showing that for parameter $\ell$ alone both problems become W[1]-hard already on $2$-degenerate graphs. Our positive result makes use of the notion of independence covering families introduced by Lokshtanov et al. Finally, we show that using such families one can obtain a simpler and unified algorithm for the standard Token Jumping Reachability problem parameterized by $k$ on both degenerate and nowhere dense classes of graphs.
翻译:当我们被给定为[G$, 2个独立设置了美元和美元, 以美元为单位, 美元为美元, 美元为美元, 美元为美元为美元, 美元为1美元为正整数。 目标是通过预先设定的重组规则来决定是否存在一个序列 $langle I_ 0, I_ 1,..., I ⁇ ell\ trangle$, 这样对于所有 $@ 0,\ldots,\ell-1美元为美元, 美元为美元, 美元为美元, 美元为美元, 美元为美元, 美元为美元, 美元为美元, 美元为美元。 我们考虑两个重新配置规则。 直观地看每套独立设置一个用于在图表的顶端端上放置的标牌。 然后, 托肯将 Onk Sliding Opilation (tal Oralal) 的结果可以显示, 当我们从一个直径到一个直径的直径, 直径显示一个直径, 我们的直到一个直径。