Rigidity, global rigidity, and graph decomposition

The recent combinatorial characterization of generic global rigidity in the plane by Jackson and Jordan (2005) [10] recalls the vital relationship between connectivity and rigidity that was first pointed out by Lovasz and Yemini (1982) [13]. The Lovasz-Yemini result states that every 6-connected graph is generically rigid in the plane, while the Jackson-Jordan result states that a graph is generically globally rigid in the plane if and only if it is 3-connected and edge-2-rigid. We examine the interplay between the connectivity properties of the connectivity matroid and the rigidity matroid of a graph and derive a number of structure theorems in this setting, some well known, some new. As a by-product we show that the class of generic rigidity matroids is not closed under 2-sum decomposition. Finally we define the configuration index of the graph and show how the structure theorems can be used to compute it.

[1]  Robert E. Tarjan,et al.  Dividing a Graph into Triconnected Components , 1973, SIAM J. Comput..

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

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

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

[5]  Brigitte Servatius,et al.  The Structure of Locally Finite Two-Connected Graphs , 1995, Electron. J. Comb..

[6]  András Frank,et al.  Connectivity and network flows , 1996 .

[7]  Brigitte Servatius,et al.  Birigidity in the Plane , 1989, SIAM J. Discret. Math..

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

[9]  Ileana Streinu,et al.  The Number of Embeddings of Minimally Rigid Graphs , 2004, Discret. Comput. Geom..

[10]  R. Graham,et al.  Handbook of Combinatorics , 1995 .

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

[12]  Audrey Lee-St. John,et al.  Pebble game algorithms and sparse graphs , 2007, Discret. Math..

[13]  W. T. Tutte.  Connectivity in graphs , 1966 .

[14]  C. St. J. A. Nash-Williams.  W. T. Tutte, Connectivity in Graphs (Toronto University Press; London: Oxford University Press), 145 pp., 42s. , 1968 .

[15]  J. Edmonds,et al.  A Combinatorial Decomposition Theory , 1980, Canadian Journal of Mathematics.

[16]  G. A. Dirac,et al.  CONNECTIVITY IN GRAPHS (Mathematical Expositions No.15) , 1968 .

[17]  Bill Jackson,et al.  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 , 2022 .

[18]  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 .

[19]  A. Recski.  Matroid theory and its applications in electric network theory and in statics , 1989 .

[20]  W. T. Tutte.  Connectivity in Matroids , 1966, Canadian Journal of Mathematics.