A Characterization of the Hamming Graphs and the Dual Polar Graphs by Completely Regular Subgraphs

In this paper we study a distance-regular graph I" of diameter d a parts per thousand yen 3 which satisfies the following two conditions: (a) For any integer i with 1 a parts per thousand currency sign i a parts per thousand currency sign d - 1 and for any pair of vertices at distance i in I" there exists a strongly closed subgraph of diameter i containing them; (b) There exists a strongly closed subgraph Delta which is completely regular in I". It is known that if Delta has diameter 1, then I" is a regular near polygon. We prove that if a strongly closed subgraph Delta of diameter j with 2 a parts per thousand currency sign j a parts per thousand currency sign d - 1 is completely regular of covering radius d - j in I", then I" is either a Hamming graph or a dual polar graph.