{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8476","name":"Who Needs Crossings? Hardness of Plane Graph Rigidity","abstract":"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.\r\n\r\nOne 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.\r\n\r\nThe 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.","keywords":["Graph Drawing","Graph Rigidity Theory","Graph Global Rigidity","Linkages","Complexity Theory","Computational Geometry"],"author":[{"@type":"Person","name":"Abel, Zachary","givenName":"Zachary","familyName":"Abel"},{"@type":"Person","name":"Demaine, Erik D.","givenName":"Erik D.","familyName":"Demaine"},{"@type":"Person","name":"Demaine, Martin L.","givenName":"Martin L.","familyName":"Demaine"},{"@type":"Person","name":"Eisenstat, Sarah","givenName":"Sarah","familyName":"Eisenstat"},{"@type":"Person","name":"Lynch, Jayson","givenName":"Jayson","familyName":"Lynch"},{"@type":"Person","name":"Schardl, Tao B.","givenName":"Tao B.","familyName":"Schardl"}],"position":3,"pageStart":"3:1","pageEnd":"3:15","dateCreated":"2016-06-10","datePublished":"2016-06-10","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Abel, Zachary","givenName":"Zachary","familyName":"Abel"},{"@type":"Person","name":"Demaine, Erik D.","givenName":"Erik D.","familyName":"Demaine"},{"@type":"Person","name":"Demaine, Martin L.","givenName":"Martin L.","familyName":"Demaine"},{"@type":"Person","name":"Eisenstat, Sarah","givenName":"Sarah","familyName":"Eisenstat"},{"@type":"Person","name":"Lynch, Jayson","givenName":"Jayson","familyName":"Lynch"},{"@type":"Person","name":"Schardl, Tao B.","givenName":"Tao B.","familyName":"Schardl"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2016.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/web.mit.edu\/tabbott\/www\/papers\/mthesis.pdf","http:\/\/www.acm.org\/pubs\/citations\/proceedings\/stoc\/62212\/p460-canny\/","http:\/\/dx.doi.org\/10.1016\/0166-218X(90)90110-X","http:\/\/arxiv.org\/abs\/math.AG\/9803150","http:\/\/arxiv.org\/abs\/math.AG\/9807023","http:\/\/dx.doi.org\/10.1006\/jcom.1999.0536","http:\/\/dx.doi.org\/10.1007\/BFb0082792","http:\/\/dx.doi.org\/10.1006\/jctb.1997.1750","https:\/\/www.cs.duke.edu\/brd\/Teaching\/Bio\/asmb\/current\/Readings3\/saxe-embeddability.pdf","http:\/\/ovid.cs.depaul.edu\/documents\/realizability.pdf"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6254","volumeNumber":51,"name":"32nd International Symposium on Computational Geometry (SoCG 2016)","dateCreated":"2016-06-10","datePublished":"2016-06-10","editor":[{"@type":"Person","name":"Fekete, S\u00e1ndor","givenName":"S\u00e1ndor","familyName":"Fekete"},{"@type":"Person","name":"Lubiw, Anna","givenName":"Anna","familyName":"Lubiw"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8476","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6254"}}}