Distance-regular Subgraphs in a Distance-regular Graph, V

Let Γ be a distance-regular graph without induced subgraphsK2,1,1andr = max {j ∣ (cj,aj,bj) = (c1,a1,b1)}. We give a necessary and sufficient condition for the existence of a strongly closed subgraph which is (cr + 1 + ar+1)-regular of diameterr + 1 containing a given pair of vertices at distancer + 1.