Even cycles and perfect matchings in planar graphs

Ear decomposition is a powerful tool for the study of the structure of matchings and enumeration of matchings. Lovász showed that a matching covered graph $G$ has an ear decomposition starting with an arbitrary edge of $G$. A graph $G$ is cycle-nice if, for each even cycle $C$ in $G$, $G-V\left(C\right)$ has a perfect matching. A matching covered graph $G$ has ear decompositions starting with an arbitrary even cycle in $G$ if and only if $G$ is a cycle-nice graph. In this paper we show that the only simple cycle-nice 3-connected planar graphs are the odd wheels and the odd prisms. Using this characterization, we show that every cycle-nice matching covered planar graph is an even cycle with multiple edges, or can be obtained from an odd wheel or an odd prism by a sequence of three types of operations.