The matching extendability of optimal $1$-embedded graphs on the projective plane

In this paper, we discuss matching extendability of optimal $1$-projective plane graphs (abbreviated as O1PPG), which are drawn on the projective plane $P^2$ so that every edge crosses another edge at most once, and has $n$ vertices and exactly $4n- 4$ edges. We first show that every O1PPG of even order is $1$-extendable. Next, we characterize $2$-extendable O1PPG's in terms of a separating cycle consisting of only non-crossing edges. Moreover, we characterize O1PPG's having connectivity exactly $5$. Using the characterization, we further identify three independent edges in those graphs that are not extendable.