Previous Up Next Article Citations From References: 0 From Reviews: 0

On characterizations of rigid graphs in the plane using spanning trees: on characterizations of rigid graphs in the plane. (English summary) Graphs Combin. 25 (2009), no. 2, 139–144.1435-5914 The generic 2-dimensional rigidity matroid is well understood, and fast algorithms to find independent sets, bases, circuits, etc., are known—see for example the chapter on planar rigidity in [J. E. Graver, B. Servatius and H. Servatius, Combinatorial rigidity, Amer. Math. Soc., Providence, RI, 1993; MR1251062 (95b:52034)]. All these algorithms are based on tree decompositions; for example, a graph G with n vertices and 2n − 2 edges is a circuit in the 2-dimensional generic rigidity matroid if and only if replacing any edge of G by any (possibly new) edge results in a graph which is the union of two spanning trees. To verify that the edge set of G is a circuit by using this characterization, (2n − 2) ( () n 2 − 1) + 1 tree decompositions are required. It is shown here that n decompositions suffice. The key to this result is the following characterization: Let e1, e2,..., e2n−2 be any circular order of edges of G. The graph G is a circuit in the generic 2-dimensional rigidity matroid if and only if G + ei − ei−1 is the union of two spanning trees for any i = 1, 2,..., 2n − 2