A distance-regular graph with bipartite geodetically closed subgraphs

Let Gamma be a distance-regular graph of diameter d, and t be an integer with 2 less than or equal to t less than or equal to d - 1 such that a(t-1) = 0. For any pair (u, v) of vertices, let Pi(u, v) be the subgraph induced by the vertices lying on shortest paths between u and v. We prove that if Pi(u, v) is a bipartite geodetically closed subgraph for some pair (u, v) of vertices at distance t, then Pi(x, y) is,a bipartite geodetically closed subgraph for any pair (x y) of vertices at distance less than or equal to t. In particular, Pi(x, y) is either a path, an ordinary polygon, a hyper cube or a projective incidence graph. (C) 2003 Elsevier Science Ltd. All rights reserved.