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Linear clique-width and modular decomposition

Robert Brignall, Michal Opler, Vincent Vatter

Abstract

A hereditary class of graphs has bounded clique-width if and only if its prime members do, but this lifting property fails for linear clique-width. We prove that a hereditary class has bounded linear clique-width if and only if its prime members do and it contains neither all quasi-threshold graphs nor all complements of quasi-threshold graphs. This generalizes a result of Brignall, Korpelainen, and Vatter, who established the result for cographs.

Linear clique-width and modular decomposition

Abstract

A hereditary class of graphs has bounded clique-width if and only if its prime members do, but this lifting property fails for linear clique-width. We prove that a hereditary class has bounded linear clique-width if and only if its prime members do and it contains neither all quasi-threshold graphs nor all complements of quasi-threshold graphs. This generalizes a result of Brignall, Korpelainen, and Vatter, who established the result for cographs.
Paper Structure (5 sections, 8 theorems, 12 equations, 1 figure)

This paper contains 5 sections, 8 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Linear Clique-Width and Universal Quasi-Threshold Graphs
  3. Modular Decomposition
  4. Proof of Theorem \ref{['thm:primes:bdd:lcw']}
  5. Concluding Remarks

Key Result

Theorem 1.1

A hereditary class of cographs has bounded linear clique-width if and only if it contains neither all quasi-threshold graphs nor the complements of all quasi-threshold graphs.

Figures (1)

  • Figure 1: The construction process for the graphs $Q_t$ and $\overline{Q}_s$ on the left and right, respectively.

Theorems & Definitions (14)

  • Theorem 1.1: Brignall, Korpelainen, and Vatter brignall:linear-clique-w:
  • Theorem 1.2
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 4 more