It is very well-known that there are precisely two minimal non-planar graphs: K5 and K3,3 (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: They each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple, 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the Möbius Ladder V10; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of V10; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of V8.
Publication Source (Journal or Book title)
Advances in Applied Mathematics
Bokal, D., Oporowski, B., Richter, R., & Salazar, G. (2016). Characterizing 2-crossing-critical graphs. Advances in Applied Mathematics, 74, 23-208. https://doi.org/10.1016/j.aam.2015.10.003