Globally Rigid Augmentation of Rigid Graphs

Abstract

. We consider the following augmentation problem: Given a rigid graph G = ( V,E ), 3 find a minimum cardinality edge set F such that the graph G (cid:48) = ( V,E ∪ F ) is globally rigid. We 4 provide a min-max theorem and a polynomial-time algorithm for this problem for several types of 5 rigidity, such as rigidity in the plane or on the cylinder. Rigidity is often characterized by some 6 sparsity properties of the underlying graph and global rigidity is characterized by redundant rigidity 7 (where the graph remains rigid after deleting an arbitrary edge) and 2-or 3-vertex-connectivity. 8 Hence, to solve the above-mentioned problem, we define and solve polynomially a combinatorial 9 optimization problem family based on these sparsity and connectivity properties. This family also 10 includes the problem of augmenting a k -tree-connected graph to a highly k -tree-connected and 2-11 connected graph. Moreover, as an interesting consequence, we give an optimal solution to the 12 so-called global rigidity pinning problem, where we aim to find a minimum cardinality vertex set X 13 for a rigid graph G = ( V,E ), such that the graph G + K X is globally rigid in R 2 where K X denotes 14 the complete graph on the vertex set X .