6 research outputs found

    Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints

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    Identifiability of data is one of the fundamental problems in data science. Mathematically it is often formulated as the identifiability of points satisfying a given set of algebraic relations. A key question then is to identify sufficient conditions for observations to guarantee the identifiability of the points. This paper proposes a new general framework for capturing the identifiability problem when a set of algebraic relations has a combinatorial structure and develops tools to analyze the impact of the underlying combinatorics on the local or global identifiability of points. Our framework is built on the language of graph rigidity, where the measurements are Euclidean distances between two points, but applicable in the generality of hypergraphs with arbitrary algebraic measurements. We establish necessary and sufficient (hyper)graph theoretical conditions for identifiability by exploiting techniques from graph rigidity theory and algebraic geometry of secant varieties

    Minimum Size Highly Redundantly Rigid Graphs in the Plane

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    A graph G is said to be k-vertex rigid in R-d if G - X is rigid in R-d for all subsets X of the vertex set of G with cardinality less than k. We determine the smallest number of edges in a k-vertex rigid graph on n vertices in R-2, for all k >= 4. We also consider k-edge-rigid graphs, defined by removing edges, as well as k-vertex globally rigid and k-edge globally rigid graphs in R-d. For d = 2 we determine the corresponding tight bounds for each of these versions, for all k >= 3. Our results complete the solutions of these extremal problems in the plane. The result on k-vertex rigidity verifies a conjecture of Kaszanitzky and Kiraly (Graphs Combin, 32:225-240, 2016). We also determine the degree of vertex redundancy of powers of cycles, with respect to rigidity in the plane, answering a question of Yu and Anderson (Int J Robust Nonlinear Control, 19(13):1427-1446, 2009)

    Connected rigidity matroids and unique realizations of graphs

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    AbstractA 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 dâ©ľ3. 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

    Birigidity in the plane

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    Abstract. We consider the 2-dimensional generic rigidity matroid R(G) of a graph G. The notions of vertex and edge birigidity are introduced. We prove that vertex birigidity of G implies the connectivity of R(G) and that the connectivity of R(G) implies the edge birigidity of G. These implications are not equivalences. A class of minimal vertex birigid graphs is exhibited and used to show that R(G) is not representable over any finite field. 1
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