Characterizing almost-median graphs II

Let M , A , S , and P be the sets of median graphs, almost-median graphs, semi-median graphs and partial cubes, respectively. Then M ⊂ A ⊂ S ⊂ P . It is proved that a partial cube is almost-median if and only if it contains no convex cycle of length greater than 4. This extends the result of Brešar (2007) [2] who proved that the same property characterizes almost-median graphs within the class of semi-median graphs.