GD 2003: Graph Drawing pp 86-97

# Noncrossing Hamiltonian Paths in Geometric Graphs

• Jakub Černý
• Zdeněk Dvořák
• Vít Jelínek
• Jan Kára
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)

## Abstract

A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: Determine a function h, where h(n) is the largest number k such that when we remove arbitrary set of k edges from a complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that $$h(n)=\Omega(\sqrt{n})$$. We also determine the function exactly in case when the removed edges form a star or a matching, and give asymptotically tight bounds in case they form a clique.

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© Springer-Verlag Berlin Heidelberg 2004

## Authors and Affiliations

• Jakub Černý
• 1
• Zdeněk Dvořák
• 1
• Vít Jelínek
• 1
• Jan Kára
• 1
1. 1.Department of Applied MathematicsCharles UniversityPraha 1