Characterization of Co-blockers for Simple Perfect Matchings in a Convex Geometric Graph

Consider the complete convex geometric graph on 2m vertices, CGG (2m), i.e., the set of all boundary edges and diagonals of a planar convex 2m-gon P. In (Keller and Perles, Israel J Math 187:465-484, 2012), the smallest sets of edges that meet all the simple perfect matchings (SPMs) in CGG(2m) (called "blockers") are characterized, and it is shown that all these sets are caterpillar graphs with a special structure, and that their total number is m. 2^m-1. In this paper we characterize the co-blockers for SPMs in CGG(2m), that is, the smallest sets of edges that meet all the blockers. We show that the co-blockers are exactly those perfect matchings M in CGG(2m) where all edges are of odd order, and two edges of M that emanate from two adjacent vertices of P never cross. In particular, while the number of SPMs and the number of blockers grow exponentially with m, the number of co-blockers grows super-exponentially.