eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-06-10
3:1
3:15
10.4230/LIPIcs.SoCG.2016.3
article
Who Needs Crossings? Hardness of Plane Graph Rigidity
Abel, Zachary
Demaine, Erik D.
Demaine, Martin L.
Eisenstat, Sarah
Lynch, Jayson
Schardl, Tao B.
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol051-socg2016/LIPIcs.SoCG.2016.3/LIPIcs.SoCG.2016.3.pdf
Graph Drawing
Graph Rigidity Theory
Graph Global Rigidity
Linkages
Complexity Theory
Computational Geometry