Untangling Polygons and Graphs

Abstract

Untangling is a process in which some vertices in a drawing 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 (n 2/3 ) vertices fixed.For any connected 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 consequence is that every 3-connected planar graph has a drawing δ such that at most O((n log n) 2/3 ) vertices are fixed in every untangling of δ.

Untangling a Planar Graph

“…They also showed that fix(C n ) ≥ √ n + 1 by applying the Erdős-Szekeres theorem to the sequence of the indices of the vertices of δ in clockwise order around some specific point. Cibulka [4] recently improved that lower bound to Ω(n 2/3 ) by applying the Erdős-Szekeres theorem not once but Θ(n 1/3 ) times.…”

“…He proved linear upper bounds on fix(G) for three-and four-connected planar graphs. Cibulka [4] gave, for any planar graph G, an upper bound on fix(G) that is a function of the number of vertices, the maximum degree, and the diameter of G. This latter bound implies, in particular, that fix(G) ∈ O((n log n) 2/3 ) for any three-connected planar graph G and that any graph H such that fix(H ) ≥ cn for some c > 0 must have a vertex of degree Ω(nc 2 / log 2 n).…”

Upper Bound Constructions for Untangling Planar Geometric Graphs

“…We thank Vida Dujmović for suggesting Cibulka's result [5] for tangling the triangulation T in the proof of Theorem 3.1; this improved the upper bound in our Corollary 3.2 from O(n .4965 ) to O(n .4948 ). We thank the anonymous referees for many useful comments and suggestions that helped improve the presentation of this paper.…”

“…Next, we describe a straight-line drawing of G. The 3-connected triangulation T * with m/2 + 2 vertices has a straight-line drawing such that in every untangling of T at most O((m log m) 2/3 ) vertices are fixed [5], and all vertices lie strictly above the x-axis. Embed the interior vertices {v 1 , .…”

“…Bose et al [3], Kang et al [13], and Ravsky and Verbitsky [15] explored drawings of several families of n-vertex planar graphs in which any untangling fixes O( √ n) vertices. Cibulka [5] showed that every 3-connected planar graph with n vertices admits a drawing for which any untangling fixes O((n log n) 2/3 ) vertices.…”