13 research outputs found
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
On the universal rigidity of generic bar frameworks
Let be a finite set. An -configuration is a mapping , where
are not contained in a proper hyper-plane. A framework in is an -configuration together with a graph such that every two points corresponding to adjacent vertices of are constrained to stay the same distance apart. A framework is said to be generic if all the coordinates of are algebraically independent over the integers. A framework in is said to be unique if there does not exist a framework in , for some , , such that for all . In this paper we present a sufficient condition for a generic framework to be unique, and we conjecture that this condition is also necessary. Connections with the closely related problems of global rigidity and dimensional rigidity are also discussed
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
Nucleation-free rigidity
When all non-edge distances of a graph realized in as a {\em
bar-and-joint framework} are generically {\em implied} by the bar (edge)
lengths, the graph is said to be {\em rigid} in . For ,
characterizing rigid graphs, determining implied non-edges and {\em dependent}
edge sets remains an elusive, long-standing open problem.
One obstacle is to determine when implied non-edges can exist without
non-trivial rigid induced subgraphs, i.e., {\em nucleations}, and how to deal
with them.
In this paper, we give general inductive construction schemes and proof
techniques to generate {\em nucleation-free graphs} (i.e., graphs without any
nucleation) with implied non-edges. As a consequence, we obtain (a) dependent
graphs in that have no nucleation; and (b) nucleation-free {\em
rigidity circuits}, i.e., minimally dependent edge sets in . It
additionally follows that true rigidity is strictly stronger than a tractable
approximation to rigidity given by Sitharam and Zhou
\cite{sitharam:zhou:tractableADG:2004}, based on an inductive combinatorial
characterization.
As an independently interesting byproduct, we obtain a new inductive
construction for independent graphs in . Currently, very few such inductive
constructions are known, in contrast to
Iterative Universal Rigidity
A bar framework determined by a finite graph and configuration in
space is universally rigid if it is rigid in any . We provide a characterization of universally rigidity for any
graph and any configuration in terms of a sequence of affine
subsets of the space of configurations. This corresponds to a facial reduction
process for closed finite dimensional convex cones.Comment: 41 pages, 12 figure
Minimally globally rigid graphs
A graph is globally rigid in if for any generic
placement of the vertices, the edge lengths
uniquely determine , up to congruence. In this
paper we consider minimally globally rigid graphs, in which the deletion of an
arbitrary edge destroys global rigidity. We prove that if is
minimally globally rigid in on at least vertices, then
. This implies that the minimum degree of
is at most . We also show that the only graph in which the upper bound on
the number of edges is attained is the complete graph . It follows
that every minimally globally rigid graph in on at least
vertices is flexible in . As a counterpart to our main result
on the sparsity of minimally globally rigid graphs, we show that in two
dimensions, dense graphs always contain nontrivial globally rigid subgraphs.
More precisely, if some graph satisfies , then
contains a subgraph on at least seven vertices that is globally rigid in
. If the well-known "sufficient connectivity conjecture" is true,
then our methods also extend to higher dimensions. Finally, we discuss a
conjectured strengthening of our main result, which states that if a pair of
vertices is linked in in , then is
globally linked in in . We prove this conjecture in the
cases, along with a variety of related results.Comment: To appear in European Journal of Combinatoric