Kombinatorikus módszerek gráfok és rúdszerkezetek merevségének vizsgálatában = Combinatorial methods in the study of rigidity of graphs and frameworks

A szerkezetek merevségi tulajdonságaira vonatkozó matematikai eredmények a statikai alkalmazásokon kívül számos más területen is hasznosíthatók. A közelmúltban sikerrel alkalmazták ezeket molekulák szerkezetének vizsgálataiban, szenzorhálózatok lokalizációs problémáiban, CAD feladatokban, stb. A kutatás célja gráfok és szerkezetek merevségi tulajdonságainak vizsgálata volt kombinatorikus módszerekkel. Igazoltuk az ú.n. Molekuláris Sejtés kétdimenziós változatát és jelentős előrelépéseket tettünk a molekuláris gráfok háromdimenziós merevségének jellemzésében is. A globálisan merev, avagy egyértelműen realizált gráfok elméletét kiterjesztettük vegyes - hossz és irány feltételeket is tartalmazó - vegyes gráfokra valamint az egyértelműen lokalizálható részekre is. Továbbfejlesztettük a szükséges gráf- és matroidelméleti módszereket. Új eredményeket értünk el a tensegrity szerkezetek, test-zsanér szerkezetek, valamint a merevség egy irányított változatával kapcsolatban is. | The mathematical theory of rigid frameworks has potential applications in various areas. It has been successfully applied - in addition to statics - in the study of flexibility of molecules, in the localization problem of sensor networks, in CAD problems, and elsewhere. In this research project we investigated the rigidity properties of graphs and frameworks by using combinatorial methods. We proved the two-dimensional version of the so-called Molecular Conjecture and made substantial progress towards a complete characterization of the rigid molecular graphs in three dimensions. We generalized the theory of globally rigid (that is, uniquely localized) graphs to mixed graphs, in which lengths as well as direction constraints are given, and to globally rigid clusters, or subgraphs. We developed new graph and matroid theoretical methods. We also obtained new results on tensegrity frameworks, body and hinge frameworks, and on a directed version of rigidity

Super Stable Tensegrities and Colin de Verdi\`{e}re Number ν\nu

A super stable tensegrity introduced by Connelly in 1982 is a globally rigid discrete structure made from stiff bars or struts connected by cables with tension. In this paper we show an exact relation between the maximum dimension that a multigraph can be realized as a super stable tensegrity and Colin de Verdi\`{e}re number~$\nu$ from spectral graph theory. As a corollary we obtain a combinatorial characterization of multigraphs that can be realized as 3-dimensional super stable tensegrities

Entangled graphs on surfaces in space

In the chemical world, as well as the physical, strands get tangled. When those strands form loops, the mathematical discipline of ‘knot theory’ can be used to analyse and describe the resultant tangles. However less has been studied about the situation when the strands branch and form entangled loops in either finite structures or infinite periodic structures. The branches and loops within the structure form a ‘graph’, and can be described by mathematical ‘graph theory’, but when graph theory concerns itself with the way that a graph can fit in space, it typically focuses on the simplest ways of doing so. Graph theory thus provides few tools for understanding graphs that are entangled beyond their simplest spatial configurations. This thesis explores this gap between knot theory and graph theory. It is focussed on the introduction of small amounts of entanglement into finite graphs embedded in space. These graphs are located on surfaces in space, and the surface is chosen to allow a limited amount of complexity. As well as limiting the types of entanglement possible, the surface simplifies the analysis of the problem – reducing a three-dimensional problem to a two-dimensional one. Through much of this thesis, the embedding surface is a torus (the surface of a doughnut) and the graph embedded on the surface is the graph of a polyhedron. Polyhedral graphs can be embedded on a sphere, but the addition of the central hole of the torus allows a certain amount of freedom for the entanglement of the edges of the graph. Entanglements of the five Platonic polyhedra (tetrahedron, octahedron, cube, dodecahedron, icosahedron) are studied in depth through their embeddings on the torus. The structures that are produced in this way are analysed in terms of their component knots and links, as well as their symmetry and energy. It is then shown that all toroidally embedded tangled polyhedral graphs are necessarily chiral, which is an important property in biochemical and other systems. These finite tangled structures can also be used to make tangled infinite periodic nets; planar repeating subgraphs within the net can be systematically replaced with a tangled version, introducing a controlled level of entanglement into the net. Finally, the analysis of entangled structures simply in terms of knots and links is shown to be deficient, as a novel form of tangling can exist which involves neither knots nor links. This new form of entanglement is known as a ravel. Different types of ravels can be localised to the immediate vicinity of a vertex, or can be spread over an arbitrarily large scope within a finite graph or periodic net. These different forms of entanglement are relevant to chemical and biochemical self-assembly, including DNA nanotechnology and metal-ligand complex crystallisation

Rigid tensegrity labelings of graphs

AbstractTensegrity frameworks are defined on a set of points in Rd and consist of bars, cables, and struts, which provide upper and/or lower bounds for the distance between their endpoints. The graph of the framework, in which edges are labeled as bars, cables, and struts, is called a tensegrity graph. It is said to be rigid in Rd if it has an infinitesimally rigid realization in Rd as a tensegrity framework. The characterization of rigid tensegrity graphs is not known for d≥2.A related problem is how to find a rigid labeling of a graph using no bars. Our main result is an efficient combinatorial algorithm for finding a rigid cable–strut labeling of a given graph in the case when d=2. The algorithm is based on a new inductive construction of redundant graphs, i.e. graphs which have a realization as a bar framework in which each bar can be deleted without increasing the degree of freedom. The labeling is constructed recursively by using labeled versions of some well-known operations on bar frameworks