In 1930, Kuratowski showed that 3,3 and 5 are the only two minor-minimal nonplanar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Širáň and Kochol ...showed that there are infinitely many k-crossing-critical graphs for any ≥2, even if restricted to simple 3-connected graphs. Recently, 2-crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such graphs. We generalize this approach to an algorithm counting Hamiltonian cycles in all 2-tiled graphs, thus extending the results of Bodroža-Pantić, Kwong, Doroslovački, and Pantić.
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