License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.SoCG.2016.3
URN: urn:nbn:de:0030-drops-58951
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Abel, Zachary ; Demaine, Erik D. ; Demaine, Martin L. ; Eisenstat, Sarah ; Lynch, Jayson ; Schardl, Tao B.

Who Needs Crossings? Hardness of Plane Graph Rigidity

LIPIcs-SoCG-2016-3.pdf (0.7 MB)


We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs, with unit-length edges and where only noncrossing configurations are considered; and unrestricted graphs (crossings allowed) with unit edge lengths (or in the global rigidity case, edge lengths in {1,2}). We show that all nine of these questions are complete for the class Exists-R, defined by the Existential Theory of the Reals, or its complement Forall-R; in particular, each problem is (co)NP-hard.

One of these nine results - that realization of unit-distance graphs is Exists-R-complete - was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NP-hard (Eades & Wormald 1990, or Cabello et al. 2007), but its membership in NP remained open; we show it is complete for the (possibly) larger class Exists-R. Global rigidity of graphs with edge lengths in {1,2} was known to be coNP-hard (Saxe 1979); we show it is Forall-R-complete.

The majority of the paper is devoted to proving an analog of Kempe's Universality Theorem - informally, "there is a linkage to sign your name" - for globally noncrossing linkages. In particular, we show that any polynomial curve phi(x,y)=0 can be traced by a noncrossing linkage, settling an open problem from 2004. More generally, we show that the nontrivial regions in the plane that may be traced by a noncrossing linkage are precisely the compact semialgebraic regions. Thus, no drawing power is lost by restricting to noncrossing linkages. We prove analogous results for matchstick linkages and unit-distance linkages as well.

BibTeX - Entry

  author =	{Zachary Abel and Erik D. Demaine and Martin L. Demaine and Sarah Eisenstat and Jayson Lynch and Tao B. Schardl},
  title =	{{Who Needs Crossingsl Hardness of Plane Graph Rigidity}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-58951},
  doi =		{10.4230/LIPIcs.SoCG.2016.3},
  annote =	{Keywords: Graph Drawing, Graph Rigidity Theory, Graph Global Rigidity, Linkages, Complexity Theory, Computational Geometry}

Keywords: Graph Drawing, Graph Rigidity Theory, Graph Global Rigidity, Linkages, Complexity Theory, Computational Geometry
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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