Counting Hamiltonian cycles in 2-tiled graphs

In 1930, Kuratowski showed that \(K_{3,3}\) and \(K_5\) are the only two minor-minimal non-planar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Šir\'{a}ň and Kochol showed that there are infinitely many \(k\)-crossing-critical graphs for any \(k\ge 2\), 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 for \(n = 2\).