A d-dimensional framework is a straight line realization of a graph G in R d . We shall only consider generic frameworks, in which the co-ordinates of all the vertices of G are algebraically independent. Two frameworks for G are equivalent if corresponding edges in the two frameworks have the same length. A framework is a unique realization of G in R d if every equivalent framework can be obtained from it by an isometry of R d . Bruce Hendrickson proved that if G has a unique realization in R d then G is (d +1)-connected and redundantly rigid. He conjectured that every realization of a (d +1)-connected and redundantly rigid graph in R d is unique. This conjecture is true for d = 1 but was disproved by Robert Connelly for d 3. We resolve the remaining open case by showing that Hendrickson's conjecture is true for d = 2. As a corollary we deduce that every realization of a 6-connected graph as a two-dimensional generic framework is a unique realization. Our proof is based on a new inductive characterization of 3-connected graphs whose rigidity matroid is connected.

LetGbe a graph andP(G, t) be the chromatic polynomial ofG. It is known thatP(G, t) has no zeros in the intervals (−∞, 0) and (0, 1). We shall show thatP(G, t) has no zeros in (1, 32/27]. In addition, we shall construct graphs whose chromatic polynomials have zeros arbitrarily close to 32/27.

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We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids.

Generalizing and unifying earlier results of W. Mader and of the second and third authors, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain min-max formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satis es local edgeconnectivity prescriptions. An extension of Edmonds' theorem on disjoint arborescences is also deduced along with a new su cient condition for the solvability of the edge-disjoint paths problem in digraphs. The approach gives rise to strongly polynomial algorithms for the corresponding optimization problems.

A 2-dimensional framework (G, p) is a graph G = (V, E) together with a map p : V → R 2 . We view (G, p) as a straight line realization of G in R 2 . Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u, v} is globally linked in G if the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G. The graph G is globally rigid if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the first two authors [12] by characterizing globally linked pairs in M -connected graphs, an important family of rigid graphs. As a by product we simplify the proof of a result of Connelly [5] which is a key step in the characterization of globally rigid graphs. We also determine the number of distinct realizations of an M -connected graph, each of which is equivalent to a given generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3].

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