Is there a name for regular bipartite graphs where each partite has the semi-completeness property (that is, for each two vertices $i$,$v$ in the partite $V$, there is a path $\{i,w,v\}$, where $w$ belongs to the other partite), but not necessarily the whole graph?
Edit: I am not looking for graphs that satisfy this property, I am wondering whether this property has a name.