Constructive approaches to the rigidity of frameworks

rigidity matroids – and show that although in dimension 2 a 1-extendable abstract rigidity matroid coincides with the generic rigidity matroid, in dimension 3 they can be different. Section 4.4 is devoted to the study of intersecting submodular functions that induce abstract rigidity matroids. We provide a necessary condition for these functions. With an additional assumption on the symmetry, we show that this necessary condition is also sufficient. We close the chapter with a discussion on a potential application of our results on abstract rigidity matroids. This work originates from the author’s master’s thesis and partly published in [85] before the enrolment to the PhD program. We include these parts in this thesis to provide a complete view of the early approach. So many details and proofs in these parts will be omitted. 4.2 Characterizing abstract rigidity matroids A vertex star of the complete graph (V,K(V )) is the set of all edges incident to some vertex v ∈ V . In [47], Graver, Servatius and Servatius posed two questions on the characterization of abstract rigidity matroids in dimension 2. Question 1 [47, page 107] Is it true that a matroid M on the edge set of the complete graph (V,K(V )) is a 2-dimensional abstract rigidity matroid if and only if all of the K4’s are circuits and all of the vertex stars minus an edge are cocircuits? Question 2 [47, page 108] Is it true that a matroid M on the edge set of the complete graph (V,K(V )) is a 2-dimensional abstract rigidity matroid if and only if all of theK4’s are circuits and r(K(U)) = 2|U |−3 for all U ⊆ V with |U | ≥ 2? Subsequently, in [48], they gave an affirmative answer to Question 1 together with its generalization in higher dimension. In [85], we show that the condition in Question 1 and the one in Question 2 are both equivalent to the property that M is an abstract rigidity matroid. This gives an affirmative answer to Question 2 and its generalization as well as an alternative proof to the result of Graver, Servatius and Servatius [48, Theorem 0.2]. As a byproduct, we obtain a polynomial algorithm for testing if a matroid given by an independence oracle is a d-dimensional abstract rigidity matroid for any fixed d. For E ⊆ K, v ∈ V \V (E) and u1, . . . , uk ∈ V (E), we call F = E+vu1+· · ·+vuk a k-valent 0-extension of E. Let Kt denote the edge set of a complete subgraph 53 4.3. 1-extendable abstract rigidity matroids on t vertices of (V,K). The principal ingredient to prove our characterization of abstract rigidity matroids is the following lemma. Lemma 4.2.1 ([85]). A matroid on the edge set of the complete graph (V,K) is a d-dimensional abstract rigidity matroid if and only if it satisfies: 1. rAd(K(V )) = d|V | − d(d+ 1)/2; 2. Each k-valent 0-extension of an independent set of Ad is also an independent set of Ad for every k ≤ d. The following theorem answers the two questions above and provides characterizations of d-dimensional abstract rigidity matroids for any d ≥ 2. Theorem 4.2.2 ([85]). The following statements are equivalent for a matroid Ad on K(V ). (i) Ad is a d-dimensional abstract rigidity matroid on K. (ii) All Kd+2’s in K(V ) are circuits of Ad and all vertex stars minus (d−1) edges are cocircuits of Ad. (iii) All Kd+2’s in K(V ) are circuits of Ad and rAd(K(U)) = d|U | − d(d + 1)/2 for every U ⊆ V with |U | ≥ d+ 1. (iv) All Kd+2’s in K(V ) are circuits of Ad and rAd(K) = d|V | − d(d+ 1)/2. Theorem 4.2.2 implies that we can discern whether a matroid Ad given by an independence oracle is a d-dimensional abstract rigidity matroid for a fixed positive integer d in polynomial time by checking condition (iv). Although it is not mentioned in [48], condition (ii) also implies a polynomial time algorithm for testing whether a given matroid is a d-dimensional abstract rigidity matroid. However, to check condition (ii), we would need to verify whether every vertex star minus (d−1) edges is a cocircuit, or, equivalently, its complement is a hyperplane, which would take O(n) calls to the independence oracle, while an algorithm using condition (iv) would need only O(n) oracle calls. 4.3 1-extendable abstract rigidity matroids In this section, extensions on a graph are regarded as extensions on its edge set. A matroidM on the edge setK of the complete graph (V,K) is called a d-dimensional 1-extendable abstract rigidity matroid if M is a d-dimensional abstract rigidity