# Hypercellular graphs: Partial cubes without Q_3^- as partial cube minor

1 ACRO - Algorithmique, Combinatoire et Recherche Opérationnelle
LIS - Laboratoire d'Informatique et Systèmes
Abstract : We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.
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Journal articles
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Cited literature [55 references]

https://hal.archives-ouvertes.fr/hal-02065775
Contributor : Victor Chepoi <>
Submitted on : Wednesday, May 20, 2020 - 12:29:22 PM
Last modification on : Wednesday, May 27, 2020 - 4:15:26 PM

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1606.02154.pdf
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• HAL Id : hal-02065775, version 1
• ARXIV : 1606.02154

### Citation

Victor Chepoi, Kolja Knauer, Tilen Marc. Hypercellular graphs: Partial cubes without Q_3^- as partial cube minor. Discrete Mathematics, Elsevier, 2020, 343 (4), pp.111678. ⟨hal-02065775⟩

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