Untangling planar graphs from a specified vertex position - Hard cases

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)[email protected]^'(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.

[1]  János Pach,et al.  Untangling a Polygon , 2001, Graph Drawing.

[2]  Oleg Verbitsky,et al.  How Much Work Does It Take To Straighten a Plane Graph Out , 2007 .

[3]  Alan Frieze,et al.  On the Length of the Longest Monotone Subsequence in a Random Permutation , 1991 .

[4]  Béla Bollobás,et al.  The Height of a Random Partial Order: Concentration of Measure , 1992 .

[5]  W. T. Tutte.  A THEOREM ON PLANAR GRAPHS , 1956 .

[6]  Douglas B. West,et al.  Vertex Degrees in Planar Graphs , 1991, Planar Graphs.

[7]  J. Baik,et al.  On the distribution of the length of the longest increasing subsequence of random permutations , 1998, math/9810105.

[8]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[9]  Xavier Goaoc,et al.  Untangling a Planar Graph , 2009, Discret. Comput. Geom..

[10]  Shaiy Pilpel,et al.  Descending subsequences of random permutations , 1990, J. Comb. Theory A.

[11]  Alexander Wolff,et al.  Moving Vertices to Make Drawings Plane , 2007, Graph Drawing.

[12]  M. Talagrand.  Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

[13]  Robin Thomas,et al.  The Four-Colour Theorem , 1997, J. Comb. Theory, Ser. B.

[14]  H. S. Wilf.  Combinatorics, Geometry and Probability: On Crossing Numbers, and some Unsolved Problems , 1997 .

[15]  Oleg Verbitsky,et al.  On Collinear Sets in Straight-Line Drawings , 2008, WG.

[16]  J. Sack,et al.  Handbook of computational geometry , 2000 .

[17]  Oleg Verbitsky,et al.  On the obfuscation complexity of planar graphs , 2007, Theor. Comput. Sci..

[18]  Alexander Wolff,et al.  Untangling a Planar Graph , 2008, SOFSEM.

[19]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[20]  Prosenjit Bose,et al.  A Polynomial Bound for Untangling Geometric Planar Graphs , 2009, Discret. Comput. Geom..

[21]  Martin Mader,et al.  Planar Graph Drawing , 2008 .

[22]  Josef Cibulka.  Untangling Polygons and Graphs , 2010, Discret. Comput. Geom..

[23]  Micha Sharir,et al.  On the number of crossing-free matchings, (cycles, and partitions) , 2006, SODA '06.

[24]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[25]  Micha Sharir,et al.  Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences , 2015, J. Comb. Theory, Ser. A.

[26]  Prosenjit Bose,et al.  A Polynomial Bound for Untangling Geometric Planar Graphs , 2007, Discret. Comput. Geom..