Bruce Hendrickson. Conditions for unique graph realizations. SIAM J. Comput., 21(1):65--84, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....rigid. A combinatorial de nition for the rigidity of G in R will be given in Section 2 of this paper. We refer the reader to [18, 19] for a formal de nition and detailed survey of the rigidity of d dimensional frameworks. The necessary condition of rigidity was strengthened by Hendrickson [9] as follows. A graph G is 2 rigid in R if deleting any edge of G results in a graph which is rigid in R . Other authors have used the terms redundantly rigid and edge birigid for 2 rigid. By using methods from di erential topology, Hendrickson proved that the 2 rigidity of G is a stronger ....

....is rigid in R . Other authors have used the terms redundantly rigid and edge birigid for 2 rigid. By using methods from di erential topology, Hendrickson proved that the 2 rigidity of G is a stronger necessary condition for the unique realizability of a generic framework (G; p) Hendrickson [9] also pointed out that the (d 1) connectivity of G is another necessary condition for a d dimensional framework (G; p) to be a unique realization of G: if G has at least d 2 vertices and has a vertex separator of size d, then we can obtain an equivalent framework to (G; p) by re ecting G along ....

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B. Hendrickson, Conditions for unique graph realizations, SIAM J. Comput. 21 (1992), no. 1, 65-84.

....the solutions. Thus arises the problem of identifying principal submatrices having a unique realization, which turns out to be NP hard [Sa79] However, several necessary conditions for unicity of realization are known, related with connectivity and generic rigidity properties of the graph pattern [Ye79, He92]. Generic rigidity of graphs can be characterized and recognized in polynomial time only in dimension k 2 ( La70] LY82] cf. survey [La97b] for more references) Call a partial matrix A a partial distance matrix if every specified principal submatrix of A is a distance matrix. Being a partial ....

B. Hendrickson. Conditions for unique graph realizations. SIAM Journal on Computing, 21:65--84, 1992.

....is fixed during the transition between solutions. Given a set of edge lengths of a graph, finding an embedding in # # which satisfies the edge length constraints, if one exists, is known to be NPcomplete [17] Determining whether there exists a unique solution may be done in polynomial time [11]. Since we incorporate a roughness term into our cost function, hence do not force precise edge lengths, none of this theory is really applicable. While this roughness term eliminates some of the solutions that minimize only the edge length component, it is still not clear whether the global ....

B. Hendrickson. Conditions for unique graph realizations. SIAM Journal of Computing, (21):65--84, 1992.

....coordinates determine the length of every edge of G. If G has no other realization with these edge lengths, up to congruence of the whole plane, then this realization is unique. A graph G is called generically globally rigid (or uniquely realizable) if any realization of G is unique. Hendrickson [5] proved that if G is generically globally rigid in the plane then G is redundantly rigid (i.e. G e is rigid for every e 2 E) and 3 connected. He conjectured that these two conditions are sucient as well. EGRES Technical Report No. 2001 08 References 15 By Laman s theorem [7] it can be seen that ....

B. Hendrickson, Conditions for unique graph realizations, SIAM J. Comput. 21 (1992), no. 1, 65-84.

....r 1 ) 2 P P 1 3 d a h 1 1 1 d 1 P P 1 3 s C r 13 C P 2 h 1 s 23 a 1 r r t, on on P 2 Q (C, r ) Figure 7: Circle defined by two incident points, P 1 and P 2 , and a tangency constraint between the circle and segment s 13 . Besides diferent concrete realizations caused by reflexions [4], each tangency constraint between circles provide two diferent solutions (see figure 2) Depending on the relative position of the centers, the given circle can be contained within the solution circle or lie outside. 4. Circle defined by an incident point and tangency constraints between the ....

B. Hendrickson. Conditions for unique graph realizations. SIAM J.Comput., 21(1):65--84, 1992.

....effective ways of combining the approach here with the techniques of distance geometry. Within the area of rigidity theory when a sparse, error free, set of distances is known the work most closely associated with problems of reconstruction is that dealing with global rigidity [4, 9, 10]. Here, the issue is to determine when a set of exact distances is sufficient to uniquely determine a point set up to congruence. Finally, we note related work of Saxe [15] proving that it is NP hard to decide whether a sufficiently sparse set of distances is consistent with a set of points in R ....

B. Hendrickson, "Conditions for unique graph realizations," SIAM J. Computing, 21(1992), pp. 65--84.

....of rigidity percolation [12, 13] Moukarzel [17] and Duxbury [18] have used a closely related approach to ours in simultaneous and independent work with a different emphasis. In x2, we review some fundamentals of rigidity theory and we recast the rigidity algorithm proposed by Hendrickson [10] in terms of a simple pebble game. Discussion of our implementation is given in x3 where the powerful method of triangularization is included which significantly improves the performance. We also discuss how to count the number of floppy modes, decompose the network into rigid clusters and locate ....

.... for determining the independence of a set of edges in two dimensions [22] Imai presented an O(n 2 ) algorithm for rigidity testing using a network flow approach [11] This complexity was matched by Gabow and Westermann using matroid sums [6] and again by Hendrickson using bipartite matching [10]. In this section we reinterpret and simplify Hendrickson s algorithm to make it practical. We will make extensive use of the following formulation of Laman s Theorem. Theorem 2.4 For a graph G = V; E) having m edges and n vertices, the following are equivalent. A. The edges of G are independent ....

Bruce Hendrickson. Conditions for unique graph realizations. SIAM J. Comput., 21(1):65--84, February 1992.

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Bruce Hendrickson. Conditions for unique graph realizations. SIAM J. Comput., 21(1):65--84, 1992.

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B. Hendrickson. Conditions for unique graph realizations. SIAM J. Comput., 21(1):65--84, February 1992.

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Hendrickson, B. Conditions for unique graph realizations. SIAM J. Comput. 21, 1 (1992), 65--84.

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B. Hendrickson, "Conditions for unique graph realizations," SIAM J. Comput., vol. 21(1), pp. 65--84, 1992.

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B. Hendrickson, "Conditions for unique graph realizations," SIAM J. Comput., vol. 21(1), pp. 65--84, 1992.

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HENDRICKSON, B. Conditions for unique graph realizations. SIAM Journal on Computing 21, 1 (1992), 65--84.

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B. Hendrickson, "Conditions for Unique Graph Realizations", SIAM J. Comput., 21 (1992), 6--84.

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B. Hendrickson. Conditions for unique graph realizations. SIAM J. Comput., 21(1):65--84, 1992.

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B. HENDRICKSON, Conditions for unique graph realizations, SIAM J. Comput., 21 (1992), pp. 65--84.

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Hendrickson B. Conditions for unique graph realizations. SIAM J. Comput., 21:65--84, 1992.

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Hendrickson, Bruce. Conditions for unique graph realizations, SIAM J. Comput., 21(1), pp. 65-84, 1992.

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