Replacing cliques by stars in quasi-median graphs

For a multi-set E of splits {bipartitions) of a finite set X, we introduce the multi-split graph G(E). This graph Is a natural extension of the Buneman graph. Indeed, it is shown that several results pertaining to the Buneman graph extend to the multi-split graph. In addition, in case E is derived from a set R of partitions of X by taking parts together with their complements, we show that the extremal instances where R is either strongly compatible or strongly incompatible are equivalent to G(E) being either a tree or a Cartesian product of star trees, respectively.