Cliquewidth and dimension

TL;DR

This work develops a unifying framework that links graph width parameters to poset dimension via Colcombet's deterministic Simon factorization. By encoding NLC-decompositions with semigroup-labelings and applying forward Ramseyan splits, the authors prove that posets of bounded cliquewidth are $ ext{dim}$-bounded, and they obtain sharp consequences for posets with bounded-treewidth cover graphs, including containment of large standard and Kelly examples. They fully characterize minor-closed graph classes for which cover-graph posets have bounded dimension, showing the critical role of excluding the cover graphs of Kelly examples. Additionally, they establish that bounded cliquewidth implies bounded Boolean dimension, with improved bounds in the treewidth regime, and provide a self-contained proof of Colcombet's theorem to underscore the methodological core. The results bridge formal language methods with structural graph theory, yielding new transduction-friendly and algorithmically relevant insights into poset dimension and 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.

Figures (1)

• Figure 1: The standard example $S_6$ (left) and $\operatorname{Kelly}_6$ (right). The poset relation goes upwards in the drawing. The pairs $(c_1,d_1),(a_2,b_2),\ldots,(a_5,b_5),(d_5,c_5)$ in $\operatorname{Kelly}_6$ induce $S_6$ as a subposet. As we can see, $\operatorname{Kelly}_k$ has a planar diagram, and the cover graph of $\operatorname{Kelly}_k$ has treewidth (and even pathwidth) at most $3$.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 8
• proof
• proof
• Lemma 11
• proof
• Claim 12