On the smallest sets blocking simple perfect matchings in a convex geometric graph

TL;DR

This paper characterizes the minimal sets of edges that block all simple perfect matchings in complete convex geometric graphs, revealing their structure as caterpillar graphs and counting their total number.

Contribution

It provides a complete structural characterization and enumeration of minimal blocking sets in convex geometric graphs, a problem previously not fully understood.

Findings

All minimal blocking sets are caterpillar graphs with a specific structure.

The total number of such sets is m * 2^{m-1}.

These results offer a complete classification of minimal blocking sets.

Abstract

In this paper we present a complete characterization of the smallest sets which block all the simple perfect matchings in a complete convex geometric graph on $2m$ vertices. In particular, we show that all these sets are caterpillar graphs with a special structure, and that their total number is $m \cdot 2^{m-1}$.