Trees in graphs of large linear cliquewidth

TL;DR

This paper extends the Pathwidth Theorem to dense graphs, showing that classes with unbounded linear cliquewidth contain large tree-like patterns that transduce all trees, advancing understanding of graph minors and transductions.

Contribution

It proves that classes of graphs with bounded cliquewidth but unbounded linear cliquewidth contain large tree-like patterns that transduce all trees, filling a key gap in graph transduction theory.

Findings

Unbounded linear cliquewidth implies presence of large tree-like patterns.

These patterns transduce all trees and subdivisions of binary trees.

Results contribute to establishing totality of cmso transduction order.

Abstract

The Pathwidth Theorem states that if a class of graphs has unbounded pathwidth, then it contains all trees as graph minors. We prove a similar result for dense graphs. More precisely, we give a finite family of tree-like patterns and prove that every graph class of bounded cliquewidth and unbounded linear cliquewidth contains arbitrarily large patterns as induced subgraphs. These patterns mso transduce all trees, and fo transduce subdivisions of all binary trees. In particular, our result provides the missing piece in establishing that the cmso transduction order is total over classes of finite graphs.