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

TL;DR

This paper characterizes hypercellular graphs, a class of partial cubes excluding a specific minor, showing they decompose into products of edges and even cycles, generalize median and cellular graphs, and relate to oriented matroids.

Contribution

It introduces hypercellular graphs, proves their structural decomposition, and links them to zonotopal complexes and median-cell properties, extending prior graph classes.

Findings

Hypercellular graphs are exactly the partial cubes with convex subgraphs built from edges and even cycles.

They generalize median and cellular graphs.

Hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids.

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.