Closed quasigeodesics, escaping from polygons, and conflict-free graph coloring
Author(s)Hesterberg, Adam Classen
Massachusetts Institute of Technology. Department of Mathematics.
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Closed quasigeodesics. A closed quasigeodesic on the surface of a polyhedron is a loop which can everywhere locally be unfolded to a straight line: thus, it's straight on faces, uniquely determined on edges, and has as much flexibility at a vertex as that vertex's curvature. On any polyhedron, at least three closed quasigeodesics are known to exist, by a nonconstructive topological proof. We present an algorithm to find one on any convex polyhedron in time O(n2[epsilon]-2- 2Ll-1 ), where [epsilon] e is the minimum curvature of a vertex, l is the length of the longest side, and t is the smallest distance within a face between a vertex and an edge not containing it. Escaping from polygons. You move continuously at speed 1 in the interior of a polygon P, trying to reach the boundary. A zombie moves continuously at speed r outside P, trying to be at the boundary when you reach it. For what r can you escape and for what r can the zombie catch you? We give exact results for some P. For general P, we give a simple approximation to within a factor of roughly 9.2504. We also give a pseudopolynomial-time approximation scheme. Finally, we prove NP-hardness and hardness of approximation results for related problems with multiple zombies and/or humans. Conflict-free graph coloring. A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. We study the natural problem of the conflict-free chromatic number XCF(G) (the smallest k for which conflict-free k-colorings exist), with a focus on planar graphs.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 56-58).
DepartmentMassachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology