Corpus ID: 220919904

Which graphs are rigid in $\ell_p^d$?

@article{Dewar2020WhichGA,
  title={Which graphs are rigid in \$\ell_p^d\$?},
  author={S. Dewar and D. Kitson and Anthony Nixon},
  journal={arXiv: Metric Geometry},
  year={2020}
}
  • S. Dewar, D. Kitson, Anthony Nixon
  • Published 2020
  • Mathematics
  • arXiv: Metric Geometry
  • We present three results which support the conjecture that a graph is minimally rigid in $d$-dimensional $\ell_p$-space, where $p\in (1,\infty)$ and $p\not=2$, if and only if it is $(d,d)$-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from $\ell_p^d$ to $\ell_p^{d+1}$. We then prove that every $(d,d)$-sparse graph with minimum degree at most $d+1$ and maximum degree at most $d+2$ is independent in $\ell_p^d… CONTINUE READING

    Figures from this paper

    References

    SHOWING 1-10 OF 25 REFERENCES
    Finite and Infinitesimal Rigidity with Polyhedral Norms
    • D. Kitson
    • Mathematics, Computer Science
    • Discret. Comput. Geom.
    • 2015
    • 18
    • PDF
    The d-dimensional rigidity matroid of sparse graphs
    • 21
    • Highly Influential
    • PDF
    Rotation in a Normed Plane
    • 3
    Infinitesimal rigidity for non-Euclidean bar-joint frameworks
    • 23
    • PDF
    Graph rigidity for unitarily invariant matrix norms
    • 7
    • PDF
    Infinitesimal Rigidity in Normed Planes
    • S. Dewar
    • Mathematics, Computer Science
    • SIAM J. Discret. Math.
    • 2020
    • 4
    • PDF
    Generating the triangulations of the projective plane
    • D. Barnette
    • Mathematics, Computer Science
    • J. Comb. Theory, Ser. B
    • 1982
    • 103
    • PDF