The k-Critical 2k-Connected Graphs for k∈{3, 4}

A noncomplete graph G is called an (n, k)-graph if it is n-connected and G−X is not (n−|X|+1)-connected for any X⊆V(G) with |X|⩽k. Mader conjectured that for k⩾3 the graph K2k+2-(1-factor) is the unique (2k, k)-graph. We settle this conjecture for k⩽4.