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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Černý, J., Dvořák, Z., Jelínek, V., Kára, J. (2004). Noncrossing Hamiltonian Paths in Geometric Graphs. In: Liotta, G. (eds) Graph Drawing. GD 2003. Lecture Notes in Computer Science, vol 2912. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24595-7_8
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DOI: https://doi.org/10.1007/978-3-540-24595-7_8
Publisher Name: Springer, Berlin, Heidelberg
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