Cliquewidth and dimension

TL;DR

This paper explores the relationship between cliquewidth, dimension, and certain subposets in posets, establishing bounds and characterizations for classes of graphs with bounded cover graph properties, and linking these to automata theory tools.

Contribution

It proves that posets with bounded cliquewidth and large dimension contain specific standard subposets, and characterizes minor-closed classes with bounded poset dimension using Kelly examples.

Findings

Posets with bounded cliquewidth and large dimension contain standard examples.

Posets with cover graphs excluding certain Kelly examples have bounded dimension.

Posets with bounded cliquewidth have bounded Boolean dimension.

Abstract

We prove that every poset with bounded cliquewidth and with sufficiently large dimension contains the standard example of dimension $k$ as a subposet. This applies in particular to posets whose cover graphs have bounded treewidth, as the cliquewidth of a poset is bounded in terms of the treewidth of the cover graph. For the latter posets, we prove a stronger statement: every such poset with sufficiently large dimension contains the Kelly example of dimension $k$ as a subposet. Using this result, we obtain a full characterization of the minor-closed graph classes $\mathcal{C}$ such that posets with cover graphs in $\mathcal{C}$ have bounded dimension: they are exactly the classes excluding the cover graph of some Kelly example. Finally, we consider a variant of poset dimension called Boolean dimension, and we prove that posets with bounded cliquewidth have bounded Boolean dimension. The proofs rely on Colcombet's deterministic version of Simon's factorization theorem, which is a fundamental tool in formal language and automata theory, and which we believe deserves a wider recognition in structural and algorithmic graph theory.