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

Consider the complete convex geometric graph on 2 m vertices, CGG ( 2 m ) , i.e., the set of all boundary edges and diagonals of a planar convex 2 m -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 ( 2 m ) (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 ( 2 m ) , 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 ( 2 m ) 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.