Bipartite embedding of trees in the plane
Discrete Applied Mathematics
Graph Drawings with no k Pairwise Crossing Edges
GD '97 Proceedings of the 5th International Symposium on Graph Drawing
Geometric Graphs with No Self-intersecting Path of Length Three
GD '02 Revised Papers from the 10th International Symposium on Graph Drawing
Topological graphs with no self-intersecting cycle of length 4
Proceedings of the nineteenth annual symposium on Computational geometry
Computational geometry column 54
ACM SIGACT News
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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 the largest number h(n) such that when we remove any set of h(n) edges from any complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that h(n)=(1/22)n. We also establish several results related to special classes of geometric graphs. Let h"1(n) denote the largest number such that when we remove edges of an arbitrary complete subgraph of size at most h"1(n) from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We prove that 12n