Hypercellular graphs: Partial cubes without Q3− as partial cube minor

Partial cubes as subdivision graphs and as generalized Petersen graphs

Isometric subgraphs of hypercubes are known as partial cubes. The subdivision graph of a graph G is obtained from G by subdividing every edge of G. It is proved that for a connected graph G its subdivision graph is a partial cube if and only if every block of G is either a cycle or a complete graph. Regular partial cubes are also considered. In particular it is shown that among the generalized ...

Partial cubes are distance graphs

Partial Cubes and Crossing Graphs

Partial cubes are defined as isometric subgraphs of hypercubes. For a partial cube G, its crossing graph G# is introduced as the graph whose vertices are the equivalence classes of the Djoković–Winkler relation Θ, two vertices being adjacent if they cross on a common cycle. It is shown that every graph is the crossing graph of some median graph and that a partial cube G is 2-connected if and on...

Partial cubes without $Q_3^-$ minors

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 Q3 (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 suc...

On Θ-graphs of partial cubes

The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K1 by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs a...