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 unitlength 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 ExistsR, defined by the Existential Theory of the Reals, or its complement ForallR; in particular, each problem is (co)NPhard.
One of these nine results  that realization of unitdistance graphs is ExistsRcomplete  was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NPhard (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 ExistsR. Global rigidity of graphs with edge lengths in {1,2} was known to be coNPhard (Saxe 1979); we show it is ForallRcomplete.
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 unitdistance linkages as well.
BibTeX  Entry
@InProceedings{abel_et_al:LIPIcs:2016:5895,
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 = {3:13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770095},
ISSN = {18688969},
year = {2016},
volume = {51},
editor = {S{\'a}ndor Fekete and Anna Lubiw},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5895},
URN = {urn:nbn:de:0030drops58951},
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 
Seminar: 

32nd International Symposium on Computational Geometry (SoCG 2016) 
Issue Date: 

2016 
Date of publication: 

09.06.2016 