A graph G=(V,E) is called k-rigid in R^{d} if |V| ≥ k+1 and after deleting at most k-1 arbitrary vertices the resulting graph is generically rigid in R^{d}. A k-rigid graph G is called minimally k-rigid if the omission of an arbitrary edge results in a graph that is not k-rigid. It was shown in [7] that the smallest possible number of edges is 2|V|-1 in a 2-rigid graph in R^{2}. We generalize this result, provide an upper bound for the number of edges of minimally 2-rigid graphs (for any d) and give examples for minimally k-rigid graphs in higher dimensions.
Bibtex entry:
@techreport{egres-12-21,
AUTHOR | = | {Kaszanitzky, Vikt{\'o}ria and Kir{\'a}ly, Csaba}, |
TITLE | = | {On minimally k-rigid graphs}, |
NOTE | = | {{\tt egres.elte.hu}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2012}, |
NUMBER | = | {TR-2012-21} |