Geodetically closed subgraphs in a distance-regular graph (Q2769879)

Let \(\Gamma\) be a distance-regular graph with \((c_1,b_1)=\cdots =(c_r,b_r)\neq (c_{r+1},b_{r+1})\), where \(r\geq 2\) and \(c_{r+1}>1\). Hiraki (1999) proved that either \(r=2\) or \((a_1,a_{r+1},c_{r+1}) =(0,0,2)\). The last case is consider in this paper. Theorem 1. Let \(\Gamma\) be a distance-regular graph of valency \(k\), \((c_1,b_1)=\cdots =(c_r,b_r)=(1,k-1)\) and \((c_{r+1},b_{r+1})=\cdots =(c_{2r},b_{2r}) =(2,k-2)\). Then \(r\leq 2\).