Blockers for Simple Hamiltonian Paths in Convex Geometric Graphs of Odd Order

Abstract

Let G be a complete convex geometric graph, and let ${\mathcal {F}}$ be a family of subgraphs of G. A blocker for ${\mathcal {F}}$ is a set of edges, of smallest possible size, that has an edge in common with every element of ${\mathcal {F}}$. In Keller and Perles (Discrete Comput Geom 60(1):1–8, 2018) 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.