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Untangling planar graphs from a specified vertex position — hard cases

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Kang, Minkyung, Pikhurko, Oleg, Ravsky, A., Schacht, M. and Verbitsky, O. (2011) Untangling planar graphs from a specified vertex position — hard cases. Discrete Applied Mathematics, Vol.159 (No.8). pp. 789-799. doi:10.1016/j.dam.2011.01.011

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Official URL: http://dx.doi.org/10.1016/j.dam.2011.01.011

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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 π of the vertex set of G into the plane. Let be the maximum integer k such that there exists a crossing-free redrawing π′ of G which keeps k vertices fixed, i.e., there exist k vertices v1,…,vk of G such that π(vi)=π′(vi) for i=1,…,k. Given a set of points X, let denote the value of minimized over π locating the vertices of G on X. The absolute minimum of is denoted by .

For the wheel graph Wn, we prove that for every X. With a somewhat worse constant factor this is also true for the fan graph Fn. We inspect also other graphs for which it is known that .

We also show that the minimum value of the parameter is always attainable by a collinear X.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Journal or Publication Title: Discrete Applied Mathematics
Publisher: Elsevier Science Ltd.
ISSN: 0166-218X
Official Date: 28 April 2011
Dates:
DateEvent
28 April 2011Published
Volume: Vol.159
Number: No.8
Page Range: pp. 789-799
DOI: 10.1016/j.dam.2011.01.011
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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