# Minimal separators in graph classes defined by small forbidden induced subgraphs

## TL;DR

This paper investigates which graph classes defined by small forbidden induced subgraphs have a polynomially bounded number of minimal separators, providing a near-complete classification for sets with up to four vertices.

## Contribution

It offers an almost complete dichotomy for classes of graphs defined by small forbidden induced subgraphs regarding their minimal separators.

## Key findings

- Identifies classes with polynomially bounded minimal separators
- Provides a near-complete classification for sets with up to four vertices
- Leaves open only two cases in the dichotomy

## Abstract

Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators. Several well-known graph classes have this property, including chordal graphs, permutation graphs, circular-arc graphs, and circle graphs. We perform a systematic study of the question which classes of graphs defined by small forbidden induced subgraphs have a polynomially bounded number of minimal separators. We focus on sets of forbidden induced subgraphs with at most four vertices and obtain an almost complete dichotomy, leaving open only two cases.