A stable cutset in a graph $G$ is a set $S\subseteq V(G)$ such that vertices of $S$ are pairwise non-adjacent and such that $G-S$ is disconnected, i.e., it is both stable (or independent) set and a cutset (or separator). Unlike general cutsets, it is $NP$-complete to determine whether a given graph $G$ has any stable cutset. Recently, Rauch et al.\ [FCT 2023] gave a number of fixed-parameter tractable (FPT) algorithms, time $f(k)\cdot |V(G)|^c$, for Stable Cutset under a variety of parameters $k$ such as the size of a (given) dominating set, the size of an odd cycle transversal, or the deletion distance to $P_5$-free graphs. Earlier works imply FPT algorithms relative to clique-width and relative to solution size. We complement these findings by giving the first results on the existence of polynomial kernelizations for \stablecutset, i.e., efficient preprocessing algorithms that return an equivalent instance of size polynomial in the parameter value. Under the standard assumption that $NP\nsubseteq coNP/poly$, we show that no polynomial kernelization is possible relative to the deletion distance to a single path, generalizing deletion distance to various graph classes, nor by the size of a (given) dominating set. We also show that under the same assumption no polynomial kernelization is possible relative to solution size, i.e., given $(G,k)$ answering whether there is a stable cutset of size at most $k$. On the positive side, we show polynomial kernelizations for parameterization by modulators to a single clique, to a cluster or a co-cluster graph, and by twin cover.

翻译：在图$G$中，稳定割集是指满足以下条件的顶点子集$S\subseteq V(G)$：$S$中的顶点两两不相邻，且$G-S$不连通，即该子集既是稳定集（独立集），又是割集（分离集）。与一般割集不同，判定给定图$G$是否存在稳定割集是$NP$完全问题。近期，Rauch等学者[FCT 2023]针对稳定割集问题提出了多种固定参数可解（FPT）算法，其时间复杂度为$f(k)\cdot |V(G)|^c$，参数$k$可选用（给定）支配集的大小、奇环横截集的大小或到无$P_5$图的删除距离等。早期研究已表明该问题在参数化为 clique-width 或解规模时存在FPT算法。本文通过首次给出稳定割集多项式核化存在性的相关结果，对上述研究进行了补充——即研究能返回参数值多项式规模等价实例的高效预处理算法。在标准假设$NP\nsubseteq coNP/poly$下，我们证明：以到单一路径的删除距离（可推广至到各类图族的删除距离）为参数，或以（给定）支配集规模为参数时，该问题不存在多项式核化。同时证明在相同假设下，以解规模为参数（即给定$(G,k)$判断是否存在规模不超过$k$的稳定割集）时也不存在多项式核化。在积极结果方面，我们给出了以到单团图、聚类图或反聚类图的调制集为参数，以及以双子覆盖为参数时的多项式核化算法。