Recognizing Shrinkable Complexes Is NP-Complete

We say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove that it is NP-complete to decide whether a (three-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes which are not shrinkable.

[1]  David Barnette,et al.  All orientable 2-manifolds have finitely many minimal triangulations , 1988 .

[2]  D. Barnette,et al.  All 2-manifolds have finitely many minimal triangulations , 1989 .

[3]  André Lieutier,et al.  Collapsing Rips complexes , 2013 .

[4]  Jovan Popovic,et al.  Progressive simplicial complexes , 1997, SIGGRAPH.

[5]  Martin Tancer.  Recognition of Collapsible Complexes is NP-Complete , 2016, Discret. Comput. Geom..

[6]  A. Fomenko,et al.  THE PROBLEM OF DISCRIMINATING ALGORITHMICALLY THE STANDARD THREE-DIMENSIONAL SPHERE , 1974 .

[7]  Vin de Silva,et al.  Topological approximation by small simplicial complexes , 2003 .

[8]  Vin de Silva,et al.  On the Local Behavior of Spaces of Natural Images , 2007, International Journal of Computer Vision.

[9]  Vin de Silva,et al.  A weak characterisation of the Delaunay triangulation , 2008 .

[10]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[11]  Xavier Goaoc,et al.  Transversal Helly numbers, pinning theorems and projection of simplicial complexes. (Nombres de Helly, théorèmes d'épinglement et projection de complexes simpliciaux) , 2011 .

[12]  Gunnar E. Carlsson,et al.  Topological estimation using witness complexes , 2004, PBG.

[13]  S. A. Lavrenchenko,et al.  Irreducible triangulations of the torus , 1990 .

[14]  André Lieutier,et al.  Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes , 2013, Comput. Geom..

[15]  Thom Sulanke.  Irreducible triangulations of low genus surfaces , 2006 .

[16]  Herbert Edelsbrunner,et al.  Topology preserving edge contraction , 1998 .

[17]  S. Melikhov,et al.  Combinatorics of embeddings , 2011, 1103.5457.

[18]  Hugues Hoppe,et al.  Progressive meshes , 1996, SIGGRAPH.

[19]  Richard Ehrenborg,et al.  The topology of the independence complex , 2006, Eur. J. Comb..

[20]  Eran Nevo.  Higher minors and van Kampen's obstruction , 2006, math/0602531.

[21]  Michael Garland,et al.  Surface simplification using quadric error metrics , 1997, SIGGRAPH.

[22]  Frédéric Chazal,et al.  A Sampling Theory for Compact Sets in Euclidean Space , 2009, Discret. Comput. Geom..

[23]  Marshall M. Cohen.  A Course in Simple-Homotopy Theory , 1973 .