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- W2079335274 abstract "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 special case when d=3. We use the independence result for even d to show that if the connectivity of G is sufficiently large in comparison to d then E has large rank in Rd(G). We use the case d=4 to show that, if G is 10-connected, then G can be made rigid in R3 by pinning down approximately three quarters of its vertices." @default.
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- W2079335274 date "2005-09-01" @default.
- W2079335274 modified "2023-09-23" @default.
- W2079335274 title "The d-dimensional rigidity matroid of sparse graphs" @default.
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- W2079335274 doi "https://doi.org/10.1016/j.jctb.2005.03.004" @default.
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