Cliques, minors and apex graphs

In this paper, we proved the following result: Let GG be a (k+2)(k+2)-connected, non-(k−3)(k−3)-apex graph where k≥2k≥2. If GG contains three kk-cliques, say L1L1, L2L2, L3L3, such that |Li∩Lj|≤k−2(1≤i