🤖 AI Summary
This study investigates deep connections between the dimension of partially ordered sets (posets) and their graph-theoretic structural parameters—specifically cliquewidth and treewidth. It addresses two central questions: (1) whether a poset of bounded cliquewidth but unbounded dimension must contain either a standard example or a Kelly example as a subposet; and (2) the precise characterization of when a poset has bounded dimension given that its cover graph has bounded treewidth. Methodologically, it establishes the first tight structural link between cliquewidth and poset dimension, integrating Colcombet’s deterministic Simon factorization, structural graph theory, formal language theory, and automata-theoretic techniques. Key contributions include: a graph-structural criterion for unbounded poset dimension; a unified resolution of the embedding problem for standard and Kelly examples; a proof that bounded cliquewidth implies bounded Boolean dimension; and a complete characterization of dimension-boundedness for posets whose cover graphs belong to specific graph classes. The work advances the convergence of structural graph theory, order theory, and formal languages.
📝 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.
