One brick at a time: a survey of inductive constructions in rigidity theory

We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding framework. We describe a number of cases in which characterisations of rigidity were proved by inductive constructions. That is, by identifying recursive operations that preserved rigidity and proving that these operations were sufficient to generate all such frameworks. We also outline the use of inductive constructions in some recent areas of particularly active interest, namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar frameworks. We summarize the key outstanding open problems related to inductions.Comment: 24 pages, 12 figures, final versio

Henneberg constructions and covers of cone-Laman graphs

We give Henneberg-type constructions for three families of sparse colored graphs arising in the rigidity theory of periodic and other forced symmetric frameworks. The proof method, which works with Laman-sparse finite covers of colored graphs highlights the connection between these sparse colored families and the well-studied matroidal (k, l)-sparse families.Comment: 14 pages, 2 figure

Global rigidity of generic frameworks on the cylinder

We show that a generic framework (G,p) on the cylinder is globally rigid if and only if G is a complete graph on at most four vertices or G is both redundantly rigid and 2-connected. To prove the theorem we also derive a new recursive construction of circuits in the simple (2,2)-sparse matroid, and a characterisation of rigidity for generic frameworks on the cylinder when a single designated vertex is allowed to move off the cylinder

Non-crossing frameworks with non-crossing reciprocals

We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudo-triangulation with one non-pointed vertex. We show that if such an embedding is sufficiently generic, then the reciprocal is non-crossing and again a pseudo-triangulation embedding of a planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation embedding of a planar Laman circuit, the reciprocal is still non-crossing and a pseudo-triangulation, but its underlying graph may not be a Laman circuit. Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal arise as the reciprocals of such, possibly singular, stresses on pseudo-triangulation embeddings of Laman circuits. All self-stresses on a planar graph correspond to liftings to piece-wise linear surfaces in 3-space. We prove characteristic geometric properties of the lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure

Global Rigidity and Symmetry of Direction-length Frameworks

PhDA two-dimensional direction-length framework (G; p) consists of a multigraph G = (V ;D;L) whose edge set is formed of \direction" edges D and \length" edges L, and a realisation p of this graph in the plane. The edges of the framework represent geometric constraints: length edges x the distance between their endvertices, whereas direction edges specify the gradient of the line through both endvertices. In this thesis, we consider two problems for direction-length frameworks. Firstly, given a framework (G; p), is it possible to nd a di erent realisation of G which satis es the same direction and length constraints but cannot be obtained by translating (G; p) in the plane, and/or rotating (G; p) by 180 ? If no other such realisation exists, we say (G; p) is globally rigid. Our main result on this topic is a characterisation of the direction-length graphs G which are globally rigid for all \generic" realisations p (where p is generic if it is algebraically independent over Q). Secondly, we consider direction-length frameworks (G; p) which are symmetric in the plane, and ask whether we can move the framework whilst preserving both the edge constraints and the symmetry of the framework. If the only possible motions of the framework are translations, we say the framework is symmetry-forced rigid. Our main result here is for frameworks with single mirror symmetry: we characterise symmetry-forced in nitesimal rigidity for such frameworks which are as generic as possible. We also obtain partial results for frameworks with rotational or dihedral symmetry.EpSRC Studentshi

Generic Rigidity Matroids with Dilworth Truncations

We prove that the linear matroid that defines generic rigidity of $d$-dimensional body-rod-bar frameworks (i.e., structures consisting of disjoint bodies and rods mutually linked by bars) can be obtained from the union of ${d+1 \choose 2}$ graphic matroids by applying variants of Dilworth truncation $n_r$ times, where $n_r$ denotes the number of rods. This leads to an alternative proof of Tay's combinatorial characterizations of generic rigidity of rod-bar frameworks and that of identified body-hinge frameworks