Metric characterization of parity graphs (Q1182856)

In a parity graph the lengths of any two induced paths between the same pair of vertices have the same parity. In this paper several characterizations of parity graphs are given, including one by forbidden isometric subgraphs, and one condition for the three distance sums \(d(u,v)+d(w,x)\), \(d(u,w)+d(v,x)\), and \(d(u,x)+d(v,w)\) between 4 vertices \(u,v,w,x\). Similar characterizations for graphs in which every block is complete or bipartite are also given.