Journal of Combinatorial Theory Series B· 2026Q1
Boolean dimension and dim-boundedness: Planar cover graph with a zero
- 2citations
- Q1SCImago
- 2026year
Short summary
Every poset with a planar cover graph and a unique minimal element has a Boolean dimension of at most 10, answering a 1989 question.
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Key points
- Posets with planar cover graphs and a unique minimal element have a Boolean dimension bounded by 10.
- This resolves a 1989 question by Nešetřil and Pudlák regarding the Boolean dimension of planar posets.
- A new reachability labeling scheme for planar digraphs with a source vertex uses O(log n) bits.
- The dimension of such posets is bounded linearly by their standard example number.
AI-generated from the title and abstract; the full text is not read.
Abstract
In 1989, Nešetřil and Pudlák posed the following challenging question: Do planar posets have bounded Boolean dimension? We show that every poset with a planar cover graph and a unique minimal element has Boolean dimension at most 10. As a consequence, we are able to show that there is a reachability labeling scheme with labels consisting of O ( log n ) bits for planar digraphs having a vertex from which every vertex is reachable. The best known scheme for general planar digraphs uses labels with O ( log 2 n ) bits (Thorup, 2004 [16] , and it remains open to determine whether a scheme using labels with O ( log n ) bits exists. The Boolean dimension result is proved in tandem with a second result showing that the dimension of a poset with a planar cover graph and a unique minimal element is bounded by a linear function of its standard example number.
The authors' abstract, as published at the source. Journal of Combinatorial Theory Series B, 2026 · DOI ↗
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Field: Computational Theory and Mathematics
Computational Theory and MathematicsComputer Science