Edge proximity and matching extension in punctured planar triangulations

A matching $M$ in a graph $G$ is said to be extendable if there exists a perfect matching of $G$ containing $M$. In 1989, it was shown that every connected planar graph with at least $8$ vertices has a matching of size three which is not extendable. In contrast, the study of extending certain matchings of size three or more has made progress in the past decade when the given graph is $5$-connected planar triangulation or $5$-connected plane graphs with few non-triangular faces.In this paper, we prove that if $G$ is a $5$-connected plane graph of even order in which at most two faces are not triangular and $M$ is a matching of size four in which the edges lie pairwise distance at least three apart, then $M$ is extendable. A related result concerning perfect matching with proscribed edges is shown as well.