Tibor Jordán

Connected rigidity matroids and unique realizations of graphs

A d-dimensional framework is a straight line realization of a graph G in Rd. 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 Rd if every equivalent framework can be obtained from it by an isometry of Rd. Bruce Hendrickson proved that if G has a unique realization in Rd then G is (d+1)-connected and redundantly rigid. He conjectured that every realization of a (d+1)-connected and redundantly rigid graph in Rd 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 Hendricksons 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.

On the optimal vertex-connectivity augmentation

Abstract This paper considers the problem of finding a minimum-cardinality set of edges for a given k -connected graph whose addition makes it ( k + 1)-connected. We give sharp lower and upper bounds for this minimum, where the gap between them is at most k − 2. This result is a generalization of the solved cases k = 1, 2, where the exact min-max formula is known. We present a polynomial-time approximation algorithm which makes a k -connected graph ( k + 1)-connected by adding a new set of edges with size at most k − 2 over the optimum.

Algorithms for Graph Rigidity and Scene Analysis

We investigate algorithmic questions and structural problems concerning graph families defined by ‘edge-counts’. Motivated by recent developments in the unique realization problem of graphs, we give an efficient algorithm to compute the rigid, redundantly rigid, M-connected, and globally rigid components of a graph. Our algorithm is based on (and also extends and simplifies) the idea of Hendrickson and Jacobs, as it uses orientations as the main algorithmic tool.

A proof of Connelly's conjecture on 3-connected circuits of the rigidity matroid

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.

Independence free graphs and vertex connectivity augmentation

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G + F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k ≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general.In this paper, we develop an algorithm which delivers an optimal solution in polynomial time for every fixed k. In the case when the size of an optimal solution is large compared to k, our algorithm is polynomial for all k. We also derive a min-max formula for the size of a smallest augmenting set in this case. A key step in our proofs is a complete solution of the augmentation problem for a new family of graphs which we call k-independence free graphs. We also prove new splitting off theorems for vertex connectivity.

Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

AbstractA two-dimensional framework (G,p) is a graph G = (V,E) together with a map p: V → ℝ2. We view (G,p) as a straight line realization of G in ℝ2. 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 %and for all equivalent frameworks (G,q), 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 [13] by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs. As a byproduct we simplify the proof of a result of Connelly [6] 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].

Edge-Connectivity Augmentation with Partition Constraints

In the well-solved edge-connectivity augmentation problem we must find a minimum cardinality set F of edges to add to a given undirected graph to make it k-edge-connected. This paper solves the generalization where every edge of F must go between two different sets of a given partition of the vertex set. A special case of this partition-constrained problem, previously unsolved, is increasing the edge-connectivity of a bipartite graph to k while preserving bipartiteness. Based on this special case we present an application of our results in statics. Our solution to the general partition-constrained problem gives a min-max formula for |F| which includes as a special case the original min-max formula of Cai and Sun [Networks, 19 (1989), pp. 151--172] for the problem without partition constraints. When k is even the min-max formula for the partition-constrained problem is a natural generalization of the unconstrained version. However, this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge. We give a strongly polynomial algorithm that solves our problem in time O(n(m + nlog n)log n). Here n and m denote the number of vertices and distinct edges of the given graph, respectively. This bound is identical to the best-known time bound for the problem without partition constraints. Our algorithm is based on the splitting off technique of Lovasz, like several known efficient algorithms for the unconstrained problem. However, unlike previous splitting algorithms, when k is odd our algorithm must handle obstacles that prevent all edges from being split off. Our algorithm is of interest even when specialized to the unconstrained problem, because it produces an asymptotically optimum number of distinct splits.

A Note on the Vertex-Connectivity Augmentation Problem

Using the polynomial algorithm given in [T. Jordan, On the optimal vertex-connectivity augmentation,J. Combin. Theory Ser. B63(1995), 8?20] ak-connected undirected graphG=(V,E) can be made (k+1)-connected by adding at mostk?2 surplus edges over (a lower bound of) the optimum. Here we introduce two new lower bounds and show that in fact the size of the solution given by (a slightly modified version of) this algorithm differs from the optimum by at most ?(k?1)/2?.

A Near Optimal Algorithm for Vertex Connectivity Augmentation

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G + F is k-vertex-connected. Polynomial algorithms for this problem are known only for k ≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. For arbitrary k, a previous result of Jordan [14] gives a polynomial algorithm which adds an augmenting set F of size at most k - 3 more than the optimum, provided G is (k - 1)-vertex-connected. In this paper we develop a polynomial algorithm which makes an l- connected graph G k-vertex-connected by adding an augmenting set of size at most ((k - l)(k - 1) + 4)=2 more than (a new lower bound for) the optimum. This extends the main results of [14,15]. We partly follow and generalize the approach of [14] and we adapt the splitting off method (which worked well on edge-connectivity augmentation problems) to vertex-connectivity. A key point in our proofs, which may also find applications elsewhere, is a new tripartite submodular inequality for the sizes of neighbour-sets in a graph.

Generic global rigidity of body-bar frameworks

A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space R^d, where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid inR^d if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G. When the configuration p is generic, redundant rigidity and (d+1)-connectivity are both necessary conditions for global rigidity. Recent results have shown that for d=2 and for generic configurations redundant rigidity and 3-connectivity are also sufficient. This gives a good combinatorial characterization in the two-dimensional case that only depends on G and can be checked in polynomial time. It appears that a similar result for d>=3 is beyond the scope of present techniques and there are examples showing that the above necessary conditions are not always sufficient. However, there is a special class of generic frameworks that have polynomial time algorithms for their generic rigidity (and redundant rigidity) in R^d for any d>=1, namely generic body-and-bar frameworks. Such frameworks are constructed from a finite number of rigid bodies that are connected by bars generically placed with respect to each body. We show that a body-and-bar framework is generically globally rigid in R^d, for any d>=1, if and only if it is redundantly rigid. As a consequence there is a deterministic polynomial time combinatorial algorithm to determine the generic global rigidity of body-and-bar frameworks in any dimension.