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

Constrained Edge-Splitting Problems

Splitting off two edges su, sv in a graph G means deleting su, sv andadding a new edge uv. Let G = (V +s,E) be k-edge-connected in V(k >= 2) and let d(s) be even. Lov´asz proved that the edges incident to scan be split off in pairs in such a way that the resulting graph on vertexset V is k-edge-connected. In this paper we investigate the existence ofsuch complete splitting sequences when the set of split edges has to meetadditional requirements. We prove structural properties of the set of thosepairs u, v of neighbours of s for which splitting off su, sv destroys k-edge-connectivity. This leads to a new method for solving problems of this type.By applying this method we obtain a short proof for a recent result ofNagamochi and Eades on planarity-preserving complete splitting sequences and prove the following new results: let G and H be two graphs on the same set V + s of vertices and suppose that their sets of edges incident to s coincide. Let G (H) be k-edge-connected (l-edge-connected, respectively) in V and let d(s) be even. Then there exists a pair su, sv which can be split off in both graphs preserving k-edge-connectivity (l-edge-connectivity, resp.) in V , provided d(s) >= 6. If k and l are both even then such a pair always exists. Using these edge-splitting results and the polymatroid intersection theorem we give a polynomial algorithm for the problem of simultaneously augmenting the edge-connectivity of two graphs by adding a (common) set of new edges of (almost) minimum size

Diszkrét optimalizálás

A jegyzet a diszkrét optimalizálás alapvető fogalmait, problémáit és algoritmikus módszereit tekinti át. Négy fejezetben tárgyalja az optimalizálási feladatokat gráfokon, az optimalizálási feladatokat matroidokon, a poliéderes kombinatorika eszköztárát, valamint kitér a merev gráfok és szerkezetek vizsgálatára is. Bemutatja a klasszikus feladatokra – gráfok párosításai, hálózati folyamok, diszjunkt utak, gráfok irányításai, legrövidebb utak, matroidok összege és metszete stb. – kidolgozott hatékony algoritmusokat és az ezekhez elvezető strukturális eredményeket. A jegyzet az ELTE TTK mesterszakos matematikus és alkalmazott matematikus hallgatói számára tartott hasonló nevű kurzus anyagának kibővített változata

Globally linked pairs of vertices in generic frameworks

A $d$-dimensional framework is a pair $(G,p)$, where $G=(V,E)$ is a graph and $p$ is a map from $V$ to $\mathbb{R}^d$. The length of an edge $xy\in E$ in $(G,p)$ is the distance between $p(x)$ and $p(y)$. A vertex pair $\{u,v\}$ of $G$ is said to be globally linked in $(G,p)$ if the distance between $p(u)$ and $p(v)$ is equal to the distance between $q(u)$ and $q(v)$ for every $d$-dimensional framework $(G,q)$ in which the corresponding edge lengths are the same as in $(G,p)$. We call $(G,p)$ globally rigid in $\mathbb{R}^d$ when each vertex pair of $G$ is globally linked in $(G,p)$. A pair $\{u,v\}$ of vertices of $G$ is said to be weakly globally linked in $G$ in $\mathbb{R}^d$ if there exists a generic framework $(G,p)$ in which $\{u,v\}$ is globally linked. In this paper we first give a sufficient condition for the weak global linkedness of a vertex pair of a $(d+1)$-connected graph $G$ in $\mathbb{R}^d$ and then show that for $d=2$ it is also necessary. We use this result to obtain a complete characterization of weakly globally linked pairs in graphs in $\mathbb{R}^2$, which gives rise to an algorithm for testing weak global linkedness in the plane in $O(|V|^2)$ time. Our methods lead to a new short proof for the characterization of globally rigid graphs in $\mathbb{R}^2$, and further results on weakly globally linked pairs and globally rigid graphs in the plane and in higher dimensions.Comment: 22 pages, 5 figure

Operations preserving the global rigidity of graphs and frameworks in the plane

AbstractA straight-line realization of (or a bar-and-joint framework on) graph G in Rd is said to be globally rigid if it is congruent to every other realization of G with the same edge lengths. A graph G is called globally rigid in Rd if every generic realization of G is globally rigid. We give an algorithm for constructing a globally rigid realization of globally rigid graphs in R2. If G is triangle-reducible, which is a subfamily of globally rigid graphs that includes Cauchy graphs as well as Grünbaum graphs, the constructed realization will also be infinitesimally rigid.Our algorithm relies on the inductive construction of globally rigid graphs which uses edge additions and one of the Henneberg operations. We also show that vertex splitting, which is another well-known operation in combinatorial rigidity, preserves global rigidity in R2

Orientations and detachments of graphs with prescribed degrees and connectivity

We give a necessary and sufficient condition for a graph to have an orientation that has k edge-disjoint arborescences rooted at a designated vertex s subject to lower and upper bounds on the in-degree at each vertex. The result is used to derive a characterization of graphs having a detachment that contains k edge-disjoint spanning trees. Efficient algorithms for finding those orientations and detachments are also described. In particular, the paper provides an algorithm for finding a connected (loopless) detachment in O(nm) time, improving on the previous best running time bound, where n and m denote the numbers of vertices and edges, respectively. © 2014 Elsevier B.V. All rights reserved