Kang, M., 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. ISSN 0166-218X

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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
Volume:	Vol.159
Number:	No.8
Page Range:	pp. 789-799
Identification Number:	10.1016/j.dam.2011.01.011
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
URI:	http://wrap.warwick.ac.uk/id/eprint/43842

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