NDLI: Untangling a Planar Graph

Content Provider	Springer Nature Link
Author	Goaoc, Xavier Kratochvíl, Jan Okamoto, Yoshio Shin, Chan Su Spillner, Andreas Wolff, Alexander
Copyright Year	2009
Abstract	A straight-line drawing δ of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G,δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-drawing problem.Further we define fix(G,δ)=n−shift(G,δ) to be the maximum number of vertices of a planar n-vertex graph G that can be fixed when untangling δ. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log\log n}$ vertices when untangling a drawing of an n-vertex graph G. If G is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$ vertices. On the other hand, we construct, for arbitrarily large n, an n-vertex planar graph G and a drawing δ G of G with $\ensuremath {\mathrm {fix}}(G,\delta_{G})\leq \sqrt{n-2}+1$ and an n-vertex outerplanar graph H and a drawing δ H of H with $\ensuremath {\mathrm {fix}}(H,\delta_{H})\leq2\sqrt{n-1}+1$ . Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.
Starting Page	542
Ending Page	569
Page Count	28
File Format	PDF
ISSN	01795376
Journal	Discrete & Computational Geometry
Volume Number	42
Issue Number	4
e-ISSN	14320444
Language	English
Publisher	Springer-Verlag
Publisher Date	2009-01-09
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Graph drawing Straight-line drawing Planarity NP-hardness Hardness of approximation Moving vertices Untangling Point-set embeddability Computational Mathematics and Numerical Analysis Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science Computational Theory and Mathematics Geometry and Topology

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Untangling a Planar Graph

Untangling Polygons and Graphs

Untangling Polygons and Graphs

Planar Embeddability of the Vertices of a Graph Using a Fixed Point Set Is NP-hard (2003)

Straight Line Triangle Representations

The point-set embeddability problem for plane graphs (2011).

Untangling a planar graph (2007)

Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices

Moving Vertices to Make Drawings Plane (2007)

Untangling a Planar Graph

Similar Documents

Untangling a Planar Graph

Untangling Polygons and Graphs

Untangling Polygons and Graphs

Planar Embeddability of the Vertices of a Graph Using a Fixed Point Set Is NP-hard (2003)

Straight Line Triangle Representations

The point-set embeddability problem for plane graphs (2011).

Untangling a planar graph (2007)

Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices

Moving Vertices to Make Drawings Plane (2007)

Untangling a Planar Graph