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Hilbert fragments

  • Csaba Biró
  • André E. Kézdy
  • Jen\Ho Lehel
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Abstract

In earlier work, we proved that the length polyhedron of an interval order has a unique Hilbert basis and that every element of this basis is a binary vector. The supports of these binary vectors form a set system whose intersection graph is a cocomparability graph. This leads naturally to the question of which graphs can arise as intersection graphs of such Hilbert-basis supports. We introduce Hilbert fragments, defined as induced subgraphs of these intersection graphs. Thus Hilbert fragments form a hereditary class of cocomparability graphs recording the intersection patterns among the supports of Hilbert-basis elements of interval-order length polyhedra. We characterize Hilbert fragments by an axiomatically defined vertex-labeling, called a track-labeling, and give applications illustrating how this characterization can reveal an underlying one-dimensional structure even when the Hilbert fragments themselves are not interval graphs.

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Cite

BibTeXRIS
BibTeX
@misc{biro2026hilbert,
  title = {{Hilbert fragments}},
  author = {Csaba Biró and André E. Kézdy and Jen\Ho Lehel},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02450},
  eprint = {2610.02450},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02450},
}
RIS
TY  - UNPB
TI  - Hilbert fragments
AU  - Csaba Biró
AU  - André E. Kézdy
AU  - Jen\Ho Lehel
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.02450
UR  - https://arxiv.org/abs/2610.02450
AB  - In earlier work, we proved that the length polyhedron of an interval order has a unique Hilbert basis and that every element of this basis is a binary vector. The supports of these binary vectors form a set system whose intersection graph is a cocomparability graph. This leads naturally to the question of which graphs can arise as intersection graphs of such Hilbert-basis supports. We introduce Hilbert fragments, defined as induced subgraphs of these intersection graphs. Thus Hilbert fragments form a hereditary class of cocomparability graphs recording the intersection patterns among the supports of Hilbert-basis elements of interval-order length polyhedra. We characterize Hilbert fragments by an axiomatically defined vertex-labeling, called a track-labeling, and give applications illustrating how this characterization can reveal an underlying one-dimensional structure even when the Hilbert fragments themselves are not interval graphs.
ER  -