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A graph G=(V,E) is called a generic cycle if |E|=2|V|-2 and every X \subset V with 2 \leq |X| \leq |V|-1 satisfies i(X) \leq 2|X|-3. Here i(X) denotes the number of edges induced by X. The operation extension subdivides an edge uw of a graph by a new vertex v and adds a new edge vz for some vertex z \not= u,w. R. Connelly conjectured that every 3-connected generic cycle can be obtained from K4 by a sequence of extensions. We prove this conjecture. As a corollary, we also obtain a special case of a conjecture of Hendrickson on generically globally rigid graphs.
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Bibtex entry:
AUTHOR | = | {Berg, Alex and Jord{\'a}n, Tibor}, |
TITLE | = | {A proof of Connelly's conjecture on 3-connected generic cycles}, |
NOTE | = | {{\tt egres.elte.hu}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2001}, |
NUMBER | = | {TR-2001-08} |