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A polynomial bound on the number of minimal separators and potential maximal cliques in $P_6$-free graphs of bounded clique number

Marcin Pilipczuk, Paweł Rzążewski

TL;DR

The paper addresses the problem of bounding the number of minimal separators and potential maximal cliques in $P_6$-free graphs with bounded clique number. It provides a concise, modular-decomposition-based argument showing that such graphs have a polynomial bound on these structures: at most $(2n)^{k+1}$ minimal separators and at most $2^{2k+2} n^{2k+3}$ PMCs. The key technical contribution is a witness-set framework using a set $Q\subseteq A$ of size at most $k$ to control separations, together with a bound on the number of connected modules, which together yield a tight polynomial bound. This establishes tameness for this graph class and has implications for polynomial-time solvability of a wide family of problems that leverage PMCs and minimal separators, while also highlighting the necessity of extra structural assumptions beyond $P_6$-freeness for tameness in related problems.

Abstract

In this note we show a polynomial bound on the number of minimal separators and potential maximal cliques in $P_6$-free graphs of bounded clique number.

A polynomial bound on the number of minimal separators and potential maximal cliques in $P_6$-free graphs of bounded clique number

TL;DR

The paper addresses the problem of bounding the number of minimal separators and potential maximal cliques in -free graphs with bounded clique number. It provides a concise, modular-decomposition-based argument showing that such graphs have a polynomial bound on these structures: at most minimal separators and at most PMCs. The key technical contribution is a witness-set framework using a set of size at most to control separations, together with a bound on the number of connected modules, which together yield a tight polynomial bound. This establishes tameness for this graph class and has implications for polynomial-time solvability of a wide family of problems that leverage PMCs and minimal separators, while also highlighting the necessity of extra structural assumptions beyond -freeness for tameness in related problems.

Abstract

In this note we show a polynomial bound on the number of minimal separators and potential maximal cliques in -free graphs of bounded clique number.
Paper Structure (2 sections, 8 theorems)

This paper contains 2 sections, 8 theorems.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

If $G$ is an $n$-vertex graph with $a$ minimal separators and $b$ potential maximal cliques, then $b \leq n(a^2 + a + 1)$ and $a \leq nb$. Furthermore, given a graph $G$, one can in time polynomial in the input and compute the list of all its minimal separators and potential maximal cliques.

Theorems & Definitions (10)

  • Theorem 1: BouchitteT01BouchitteT02
  • Theorem 2: BouchitteT01FominTV15, informal statement
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Corollary 6
  • proof
  • Lemma 7: cf. Lemma 4.2 of GrzesikKPP22
  • Lemma 8
  • proof