Weak-geodetically closed subgraphs in distance-regular graphs

Let Γ = (X, R) denote a distance-regular graph with diameter D ≥ 2 and distance function δ. A (vertex) subgraph Ω ⊆ X is said to be weak-geodetically closed whenever for all x, y ∈ Ω and all z ∈ X, δ(x, z) + δ(z, y) ≤ δ(x, y) + 1 → z ∈ Ω. We show that if the intersection number c₂ > 1 then any weak-geodetically closed subgraph of X is distance-regular. Γ is said to be i-bounded, whenever for all x, y ∈ X at distance δ(x, y) ≤ i, x, y are contained in a common weak-geodetically closed subgraph of Γ of diameter δ(x, y). By a parallelogram of length i, we mean a 4-tuple xyzw of vertices in X such that δ(x, y) = δ(z, w) = 1, δ(x, w) = i, and δ(x, z) = δ(y, z) = δ(y, w) = i - 1. We prove the following two theorems.