A characterization of 1-cycle resonant graphs among bipartite 2-connected plane graphs

It is proved that a bipartite 2-connected plane graph in which the common boundary of adjacent faces is a simple curve is 1-cycle resonant if and only if the outer face of GG is alternating and each inner vertex has degree two. This extends a result from [X. Guo, F. Zhang, kk-cycle resonant graphs, Discrete Math. 135 (1994) 113–20] that a hexagonal system is 1-cycle resonant if and only if it is catacondensed.