What is the meaning of the second, third and fourth graph? The image is from arXiv:hep-th/9912072.
Interestingly, non-planar graphs are suppressed due to extra phase factors. (This is somewhat similar to the planar limit/large-$N_c$-expansion in $SU(N_c)$ Yang-Mills theory, although the non-planar suppression is there caused because non-planar diagrams have fewer traces.)
Returning to OP's question, the second, third and fourth graph in Fig. 7 are non-planar graphs (again due to that the cyclic order of legs in vertices matters).
- S. Minwalla, M. Van Raamsdonk & N. Seiberg, Non-commutative Perturbative Dynamics, arXiv:hep-th/9912072.