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

Control of Formations with Non-rigid and Hybrid Graphs

This thesis studies the problem of control of multi-agent formations, of which the interaction architectures can be modeled by undirected and directed graphs or a mixture of the two (hybrid graphs). The algorithms proposed in this thesis can be applied to control the architectures of multi-agent systems or sensor networks, and the developed control laws can be employed in the autonomous agents of various types within multi-agent systems. This thesis discusses two major issues. The first tackles formations with undirected and directed underlying graphs, more specifically, the problems of rigidity restoration and persistence verification for multi-agent formations are studied. The second discusses the control of formations with both undirected and directed interaction architectures (hybrid formations) by distance-based control methods. The main contributions of this thesis are: definition of spindle agent and basic graphs for non-rigid undirected graphs, development of new operations for the constructions of undirected and directed graphs, design of graph rigidity restoration strategy by merging two or more non-rigid graphs, development of new persistence analysis strategy for arbitrary directed graphs, definition and investigation of hybrid formations and the underlying hybrid graphs, verification of persistence and minimal persistence for hybrid graphs, as well as the control of persistent hybrid formations by distance-based approaches