There Exist Highly Critically Connected Graphs of Diameter Three

Let κ(G) denote the (vertex) connectivity of a graph G. For ℓ≥0, a noncomplete graph of finite connectivity is called ℓ-critical if κ(G−X)=κ(G)−|X| for every X⊆V(G) with |X|≤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.