The d-dimensional rigidity matroid of sparse graphs

@article{Jackson2005TheDR,
  title={The d-dimensional rigidity matroid of sparse graphs},
  author={Bill Jackson and Tibor Jord{\'a}n},
  journal={J. Comb. Theory, Ser. B},
  year={2005},
  volume={95},
  pages={118-133}
}
Let Rd(G) be the d-dimensional rigidity matroid for a graph G = (V, E). For X ⊆ V let i(X) be the number of edges in the subgraph of G induced by X. We derive a min-max formula which determines the rank function in Rd(G) when G has maximum degree at most d+ 2 and minimum degree at most d+1. We also show that if d is even and i(X) ≤ 12 [(d+2)|X|− (2d+2)] for all X ⊆ V with |X| ≥ 2 then E is independent in Rd(G). We conjecture that the latter result holds for all d ≥ 2 and prove this for the… CONTINUE READING

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