Egerváry Research Group on Combinatorial Optimization Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs 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].

[1]  Toshihide Ibaraki,et al.  A linear-time algorithm for finding a sparsek-connected spanning subgraph of ak-connected graph , 1992, Algorithmica.

[2]  L. Lovász,et al.  On Generic Rigidity in the Plane , 1982 .

[3]  R. Connelly Rigidity and energy , 1982 .

[4]  A. Savvides,et al.  Network localization in partially localizable networks , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[5]  Bill Jackson,et al.  Egerváry Research Group on Combinatorial Optimization Connected Rigidity Matroids and Unique Realizations of Graphs Connected Rigidity Matroids and Unique Realizations of Graphs , 2022 .

[6]  G. Laman On graphs and rigidity of plane skeletal structures , 1970 .

[7]  Tibor Jordán,et al.  Algorithms for Graph Rigidity and Scene Analysis , 2003, ESA.

[8]  Robert Connelly,et al.  Generic Global Rigidity , 2005, Discret. Comput. Geom..

[9]  Brian D. O. Anderson,et al.  Rigidity, computation, and randomization in network localization , 2004, IEEE INFOCOM 2004.

[10]  Walter Whiteley,et al.  Some matroids from discrete applied geometry , 1996 .

[11]  Bruce Hendrickson,et al.  Conditions for Unique Graph Realizations , 1992, SIAM J. Comput..

[12]  James G. Oxley,et al.  Matroid theory , 1992 .

[13]  H. Gluck Almost all simply connected closed surfaces are rigid , 1975 .

[14]  Lebrecht Henneberg,et al.  Die graphische Statik der Starren Systeme , 1911 .

[15]  Tibor Jordán,et al.  A proof of Connelly's conjecture on 3-connected circuits of the rigidity matroid , 2003, J. Comb. Theory, Ser. B.

[16]  Ileana Streinu,et al.  The Number of Embeddings of Minimally Rigid Graphs , 2002, SCG '02.