A graph G = (V, E) is called a generic circuit if E = 2 V - 2 and every X ⊂ V with 2≤ X ≤ V - 1 satisfies i(X)≤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 ≠ u, w. Connelly conjectured that every 3-connected generic circuit 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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics