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Untangling polygons and graphs (0802.1312v2)
Published 10 Feb 2008 in cs.CG and cs.DM
Abstract: Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at least \Omega(n{2/3}) vertices fixed. For any graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound O((n log n){2/3}) for all 3-vertex-connected planar graphs.