Abstract
Gluck has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K 5-minor freeness guarantees the stress freeness. More generally, we prove that every K r+2-minor free graph is generically r-stress free for 1≤r≤4. (This assertion is false for r≥6.) Some further extensions are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Asimov and B. Roth: The rigidity of graphs, Trans. Amer. Math. Soc. 245 (1978), 279–289.
L. Asimov and B. Roth: The rigidity of graphs: part II; J. Math. Anal. Appl. 68 (1979), 171–190.
L. Cauchy: Sur les polygones et les polyèdres, Second Memoire, I. Ecole Polytechnique 245 (1813) (= Oeuvres complètes d’Augustin Cauchy 2nd sér., Tome 1 (1905), pp. 26-38).
Y. Colin de Verdière: Sur un nouvel invariant des graphes et un critère de planarité, J. Comb. Th., Ser. B 50 (1990), 11–21. (English translation: On a new graph invariant and a criterion for planarity, in Graph Structure Theory (N. Robertson and P. Seymour eds.), Contemp. Math., A.M.S., (1993), pp. 137–147.).
R. Diestel: Graph Theory, Second edition, Springer-Verlag, N.Y., (2000).
H. Gluck: Almost all simply connected closed surfaces are rigid, in Geometric topology; Lecture Notes in Math., Vol. 438, pp. 225–239 (1975).
G. Kalai: Rigidity and the lower bound theorem, Inven. Math. 88 (1987), 125–151.
G. Kalai: Algebraic Shifting, Advanced Studies in Pure Math. 33 (2002), 121–163.
K. Kuratowski: Sur le probléme des courbes gauches en topologie, Fund. Math. 15 (1930), 271–283.
K. W. Lee: Generalized stress and motion, in Polytopes: Abstract, Convex and Computational (T. Briztriczky eds.), (1995), 249–271.
L. Lovász and A. Schrijver: A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs, Proc. Amer. Math. Soc. 126 (1998), 1275–1285.
W. Mader: Homomorphiesätze für Graphen, Math. Ann. 178 (1968), 154–168.
W. Mader: 3n–5 edges do force a subdivision of K 5, Combinatorica 18(4) (1998), 569–595.
E. Nevo: Embeddability and stresses of graphs, arXiv: math.CO/0411009.
N. Robertson, P. D. Seymour and R. Thomas: Linkless embeddings of graphs in 3-space, Bull. Amer. Math. Soc. 28 (1993), 84–89.
Z. Song: The extremal function for K −8 minors, J. Comb. Th., Ser. B 95(2) (2005), 300–317.
E. Steinitz and H. Rademacher: Volgesungen über die Theorie der Polyeder, Berlin-Göttingen, Springer, 1934.
W. Whiteley: Vertex splitting in isostatic frameworks, Struc. Top. 16 (1989), 23–30.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by an I.S.F. grant.