There exist highly critically connected graphs of diameter three

Let kappa(G) denote the (vertex) connectivity of a graph G. For l >= 0, a noncomplete graph of finite connectivity is called l-critical if kappa(G-X) = kappa(G)-vertical bar X vertical bar for every X subset of V(G) with vertical bar X vertical bar <= l. Mader proved that every 3-critical graph has diameter at most 4 and asked for 3-critical graphs having diameter exceeding 2. Here we give an affirmative answer by constructing an l-critical graph of diameter 3 for every l >= 3.