On polynomial kernelization for Stable Cutset

TL;DR

This work initiates a systematic study of polynomial kernelizations for the NP-hard Stable Cutset problem. It delivers both positive and negative results: polynomial kernels are obtained when parameterizing by modulators to cluster, co-cluster, and twin-cover graph classes, with kernel sizes ranging from $O(|X|^3)$ to $O(|X|^5)$, while strong non-kernelization results rule out polynomial kernels for parameterizations like distance to a single path, domination number, and solution size under standard complexity assumptions. The results leverage marking-based reductions, modular decomposition concepts, and MK[2]-hardness reductions from Set Splitting and related problems to establish tight boundaries for kernelization. Collectively, the paper clarifies which structural graph parameters enable preprocessing to polynomial size and identifies key avenues for further exploration (e.g., other modulators such as to cographs).

Abstract

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.

Figures (1)

• Figure 1: The graph $G$ from the set splitting instance $(U,{\cal F})$ with $U=\{w,x,y,z\}$, ${\cal F}=\{C,D\}$, $C=\{x,y\}$ and $D=\{w,x,z\}$.

Theorems & Definitions (56)

• Theorem 1
• Theorem 2
• Proposition 1
• Proposition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof