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The polynomial characterization of tope graphs of the lopsided sets

  • Xuan Zheng
  • Yan-Ting Xie
  • Shou-Jun Xu
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Abstract

The cube polynomial CG(x) generates the number of k-cubes on a graph G. As a subclass of partial cubes, the tope graphs of lopsided sets (LOPs) generalize daisy cubes and median graphs. In this paper, we prove that every tope graph of a LOP shares its cube polynomial with some daisy cube, thereby answering affirmatively a problem posed earlier by the authors. Furthermore, we present explicit expressions for the cube polynomials of tope graphs of LOPs: CG(x)=fK(x+1), where fK(x) is the f-polynomial of the cubical complex 𝒦 of G. Finally, we show that a class 𝒒 of partial cubes is the class of tope graphs of LOPs if and only if the following equivalent conditions hold: (a) 𝒒 is the maximal pc-minor-closed class such that every graph in 𝒒 has the same cube polynomial as some daisy cube; (b) for every Gβˆˆπ’’, each antipodal subgraph of G has the same cube polynomial as some daisy cube.

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Cite

BibTeXRIS
BibTeX
@misc{zheng2026polynomial,
  title = {{The polynomial characterization of tope graphs of the lopsided sets}},
  author = {Xuan Zheng and Yan-Ting Xie and Shou-Jun Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22741},
  eprint = {2609.22741},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22741},
}
RIS
TY  - UNPB
TI  - The polynomial characterization of tope graphs of the lopsided sets
AU  - Xuan Zheng
AU  - Yan-Ting Xie
AU  - Shou-Jun Xu
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22741
UR  - https://arxiv.org/abs/2609.22741
AB  - The cube polynomial C_G(x) generates the number of k-cubes on a graph G. As a subclass of partial cubes, the tope graphs of lopsided sets (LOPs) generalize daisy cubes and median graphs. In this paper, we prove that every tope graph of a LOP shares its cube polynomial with some daisy cube, thereby answering affirmatively a problem posed earlier by the authors. Furthermore, we present explicit expressions for the cube polynomials of tope graphs of LOPs: C_G(x)=f_K(x+1), where f_K(x) is the f-polynomial of the cubical complex 𝒦 of G. Finally, we show that a class 𝒒 of partial cubes is the class of tope graphs of LOPs if and only if the following equivalent conditions hold: (a) 𝒒 is the maximal pc-minor-closed class such that every graph in 𝒒 has the same cube polynomial as some daisy cube; (b) for every Gβˆˆπ’’, each antipodal subgraph of G has the same cube polynomial as some daisy cube.
ER  -