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

TL;DR

This paper characterizes the minimal edge sets (blockers) that intersect all simple Hamiltonian paths in convex geometric graphs of odd order, revealing more complex structures than in the even case.

Contribution

It provides an explicit description of blockers for simple Hamiltonian paths in odd-order convex geometric graphs, extending previous work from the even case.

Findings

Blockers in odd-order graphs are more complex than in even-order graphs.

Blockers are not necessarily simple structures in the odd case.

The proof of the characterization is more involved due to increased complexity.

Abstract

Let G be a complete convex geometric graph, and let F be a family of subgraphs of G. A blocker for F is a set of edges, of smallest possible size, that has an edge in common with every element of F. In [C. Keller and M. A. Perles, Blockers for simple Hamiltonian paths in convex geometric graphs of even order, Disc. Comput. Geom., 60(1) (2018), pp. 1-8] we gave an explicit description of all blockers for the family of simple (i.e., non-crossing) Hamiltonian paths (SHPs) in G in the `even' case |V(G)|=2m. It turned out that all the blockers are simple caterpillar trees of a certain class. In this paper we give an explicit description of all blockers for the family of SHPs in the `odd' case |V(G)|=2m-1. In this case, the structure of the blockers is more complex, and in particular, they are not necessarily simple. Correspondingly, the proof is more complicated.