Finding a Maximum Minimal Separator: Graph Classes and Fixed-Parameter Tractability

TL;DR

This paper investigates the computational complexity of finding maximum minimal separators in various graph classes, establishing NP-hardness results and exploring fixed-parameter tractability with algorithms and lower bounds.

Contribution

It proves NP-hardness for planar bipartite, co-bipartite, and line graphs, and provides a fixed-parameter algorithm with bounds, along with ETH-based complexity insights.

Findings

NP-hardness in planar bipartite graphs

Fixed-parameter algorithm with $2^{O(k)}n^{O(1)}$ runtime

No subexponential parameterized algorithm unless ETH fails

Abstract

We study the problem of finding a maximum cardinality minimal separator of a graph. This problem is known to be NP-hard even for bipartite graphs. In this paper, we strengthen this hardness by showing that for planar bipartite graphs, the problem remains NP-hard. Moreover, for co-bipartite graphs and for line graphs, the problem also remains NP-hard. On the positive side, we give an algorithm deciding whether an input graph has a minimal separator of size at least $k$ that runs in time $2^{O(k)}n^{O(1)}$. We further show that a subexponential parameterized algorithm does not exist unless the Exponential Time Hypothesis (ETH) fails. Finally, we discuss a lower bound for polynomial kernelizations of this problem.