Graphs of Acyclic Cubical Complexes

It is well known that chordal graphs are exactly the graphs of acyclic simplicial complexes. In this note we consider the analogous class of graphs associated with acyclic cubical complexes. These graphs can be characterized among median graphs by forbidden convex subgraphs. They possess a number of properties (in analogy to chordal graphs) not shared by all median graphs. In particular, we disprove a conjecture of Mulder on star contraction of median graphs. A restricted class of cubical complexes for which this conjecture would hold true is related to perfect graphs.