Blockers for simple Hamiltonian paths in convex geometric graphs of even order

TL;DR

This paper demonstrates that the minimal edge sets blocking all simple Hamiltonian paths in a complete convex geometric graph of even order are identical to those blocking all simple perfect matchings, simplifying previous proofs.

Contribution

It establishes that the blockers for simple Hamiltonian paths are the same as for simple perfect matchings in such graphs, providing a simpler proof.

Findings

Blockers for SHPs are identical to those for SPMs.

The proof of this equivalence is shorter and simpler than previous work.

The result applies to complete convex geometric graphs of even order.

Abstract

Let G be a complete convex geometric graph on 2m vertices, and let F be a family of subgraphs of G. A blocker for F is a set of edges, of smallest possible size, that meets every element of F. In [C. Keller and M. A. Perles, On the smallest sets blocking simple perfect matchings in a convex geometric graph, Israel J. Math. 187 (2012), pp. 465-484], we gave an explicit description of all blockers for the family of simple perfect matchings (SPMs) of G. In this paper we show that the family of simple Hamiltonian paths (SHPs) in G has exactly the same blockers as the family of SPMs. Our argument is rather short, and provides a much simpler proof of the result of [KP12].