Many generalizations of the well-known class of median graphs have been studied so far. Most of them share a characterization in terms of a certain gated amalgamation procedure from Cartesian products of the graphs that belong to a special subclass of the class, i.e., the subclass of its prime graphs. We investigated such class of graphs in which the prime graphs are all connected bridged graphs without cut-vertices. In this talk it will be presented that these graphs are precisely the retracts of Cartesian products of bridged graphs and, in turn, they are also precisely the weakly modular graphs that do not contain $K_{2,3},$ wheels $W_4,~W_5$ and 4-wheel minus one edge $W_4^-$ as induced subgraphs. This is joint work with Bo\v stjan Bre\v sar, Jeremie Chalopin and Victor Chepoi.