Modular graph

[2] It is not possible for a modular graph to contain a cycle of odd length.

For, if C is a shortest odd cycle in a graph, x is a vertex of C, and yz is the edge of C farthest from x, there could be no median m(x, y, z).

In this case, the only vertices on the shortest path yz are y and z themselves.

Neither can belong to a shortest path from x to the other without shortcutting C and creating a shorter odd cycle.

[1] The modular graphs contain as a special case the median graphs, in which every triple of vertices has a unique median;[1] median graphs are related to distributive lattices in the same way that modular graphs are related to modular lattices.