Linear clique-width and modular decomposition

TL;DR

This paper characterizes when hereditary graph classes have bounded linear clique-width, showing it depends on prime members and exclusion of certain graph classes, generalizing previous results for cographs.

Contribution

It provides a complete characterization of hereditary classes with bounded linear clique-width, extending known results beyond cographs.

Findings

Hereditary class has bounded linear clique-width iff prime members do and excludes all quasi-threshold graphs and their complements.

Generalizes prior results from cographs to broader classes.

Establishes a new criterion linking prime graphs and linear clique-width.

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.