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dc.contributor.authorAbel, Zachary R
dc.contributor.authorDemaine, Erik D
dc.contributor.authorDemaine, Martin L
dc.contributor.authorEisenstat, Sarah Charmian
dc.contributor.authorLynch, Jayson R.
dc.contributor.authorSchardl, Tao Benjamin
dc.date.accessioned2017-10-24T18:10:09Z
dc.date.available2017-10-24T18:10:09Z
dc.date.issued2016-06
dc.identifier.isbn978-3-95977-009-5
dc.identifier.issn1868-8969
dc.identifier.urihttp://hdl.handle.net/1721.1/111968
dc.description.abstractWe 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.en_US
dc.language.isoen_US
dc.publisherDagstuhl Publishingen_US
dc.relation.isversionofhttp://dx.doi.org/10.4230/LIPIcs.SoCG.2016.3en_US
dc.rightsCreative Commons Attribution 4.0 International Licenseen_US
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/en_US
dc.sourceDagstuhl Publishingen_US
dc.titleWho Needs Crossings? Hardness of Plane Graph Rigidityen_US
dc.typeArticleen_US
dc.identifier.citationAbel, Zachary et al. "Who Needs Crossings? Hardness of Plane Graph Rigidity." 32nd International Symposium on Computational Geometry (SoCG 2016), June 14-17 2016, Boston, Massachusetts, USA , edited by Sandor Fekete and Anna Lubiw, Dagstuhl Publishing, June 2016 © Zachary Abel, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Jayson Lynch, and Tao B. Schardlen_US
dc.contributor.departmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratoryen_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.contributor.mitauthorAbel, Zachary R
dc.contributor.mitauthorDemaine, Erik D
dc.contributor.mitauthorDemaine, Martin L
dc.contributor.mitauthorEisenstat, Sarah Charmian
dc.contributor.mitauthorLynch, Jayson R.
dc.contributor.mitauthorSchardl, Tao Benjamin
dc.relation.journal32nd International Symposium on Computational Geometry (SoCG 2016)en_US
dc.eprint.versionFinal published versionen_US
dc.type.urihttp://purl.org/eprint/type/ConferencePaperen_US
eprint.statushttp://purl.org/eprint/status/NonPeerRevieweden_US
dspace.orderedauthorsAbel, Zachary; Demaine, Erik D.; Demaine, Martin L.; Eisenstat, Sarah; Lynch, Jayson; Schardl, Tao B.en_US
dspace.embargo.termsNen_US
dc.identifier.orcidhttps://orcid.org/0000-0002-4295-1117
dc.identifier.orcidhttps://orcid.org/0000-0003-3803-5703
dc.identifier.orcidhttps://orcid.org/0000-0002-3182-1675
dc.identifier.orcidhttps://orcid.org/0000-0003-0198-3283
mit.licensePUBLISHER_CCen_US


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