Axiomatic characterization of the interval function of partial cubes and partial Hamming 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...

Tree-like partial Hamming graphs

Tree-like partial cubes were introduced in [B. Brešar, W. Imrich, S. Klavžar, Tree-like isometric subgraphs of hypercubes, Discuss. Math. Graph Theory, 23 (2003), 227–240] as a generalization of median graphs. We present some incorrectnesses from that article. In particular we point to a gap in the proof of the theorem about the dismantlability of the cube graph of a tree-like partial cube and ...

Partial cubes are distance graphs

Partial Hamming graphs and expansion procedures

Structural properties of isometric subgraphs of Hamming graphs are presented, generalizing certain results on quasi-median graphs. Consequently, a relation on the edge set of a graph which is closely related to Winkler–Djokovi6 c’s relation is introduced and used for a characterization of isometric subgraphs of Hamming graphs. Moreover, some results considering semi-median graphs and expansions...

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...