Untangling planar graphs from a specified vertex position-Hard cases

  • Authors:

  • M. Kang;O. Pikhurko;A. Ravsky;M. Schacht;O. Verbitsky

  • Affiliations:

  • Institut für Informatik, Humboldt Universität zu Berlin, D-10099 Berlin, Germany;Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States;Institute for Applied Problems of Mechanics and Mathematics, Naukova St.3, Lviv 79060, Ukraine;Institut für Informatik, Humboldt Universität zu Berlin, D-10099 Berlin, Germany;Institute for Applied Problems of Mechanics and Mathematics, Naukova St.3, Lviv 79060, Ukraine

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2011

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Abstract

Given a planar graph G, we consider drawings of G in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding @p of the vertex set of G into the plane. Let fix(G,@p) be the maximum integer k such that there exists a crossing-free redrawing @p^' of G which keeps k vertices fixed, i.e., there exist k vertices v"1,...,v"k of G such that @p(v"i)=@p^'(v"i) for i=1,...,k. Given a set of points X, let fix^X(G) denote the value of fix(G,@p) minimized over @p locating the vertices of G on X. The absolute minimum of fix(G,@p) is denoted by fix(G). For the wheel graph W"n, we prove that fix^X(W"n)@?(2+o(1))n for every X. With a somewhat worse constant factor this is also true for the fan graph F"n. We inspect also other graphs for which it is known that fix(G)=O(n). We also show that the minimum value fix(G) of the parameter fix^X(G) is always attainable by a collinear X.