András Recski

A network theory approach to the rigidity of skeletal structures part II. Laman’s theorem and topological formulae

In the first part of the paper [8] we associated matroidal models to skeletal structures. This modelling, where the matroids were defined on the set of velocities of the joints, enabled us to characterize generic rigidity in terms of what we called the Master Matroid. Furthermore, following the way of thinking in network theory, we could formulate the matroids of interconnected structures in terms of matroid union. In the present second part we study the relation of our characterization of generic rigidity to Laman’s theorem. As a byproduct, we obtain the analogues of the so called topological formulae of linear network theory. For terminology, notation etc. the reader is referred to (81.

Rigid tensegrity labelings of graphs

Tensegrity frameworks are defined on a set of points in R^d 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 R^d if it has an infinitesimally rigid realization in R^d 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.

Minimax Results and Polynomial Algorithms in VLSI Routing

This survey is an informal introduction to some results in VLSI routing. There are hundreds or perhaps thousands of papers on this broad subject, including heuristic algorithms with sometimes very good practical performance as well as some very deep theoretical results. The present article tries to give an introduction for interested mathematicians, concentrating on some simple special subproblems which can be used as illustrations of the applicability of combinatorial optimization.

Network theory and transversal matroids

Abstract Various matroidal models for the solvability of active linear networks are described in a unified way and compared with one another. One of the conclusions is that different authors used the concept of “generality” in different ways and their methods are not equivalent. Another conclusion is that some of the seemingly equivalent data structures can be better for storing the models of the network devices in a computer. Finally, a hierarchy of network devices is introduced from this point of view. The classes of this hierarchy are proved to be characterized by transversal and fundamental transversal matroids.

Unconstrained multilayer switchbox routing

Consider the gradually more and more complexproblems of single row routing, channel routing and switchbox routing on the one hand, and the gradually less and less restrictivemodels (1-layer, Manhattan, unconstrained 2-layer, multilayer) on the other hand. The single row routing problems can always be solved in the Manhattan model, and the channel routing problem can always be solved in the unconstrained 2-layer model, in fact, both in linear time. In this paper, we show that the switchbox routing problem is solvable in the multilayer model, also in linear time.

On the sum of matroids, III

Almost irreducible graphic matroids are characterized and the structure of all the possible decompositions of a graphic matroid @? into the sum of matroids is described (provided that @? is connected and not a single circuit). Some contributions to the solvability of the matroid equation %plane1D;49C; @? @? = @? are also obtained.

Combinatorial Conditions for the Rigidity of Tensegrity Frameworks

Rigidity of bar-and-joint frameworks has been studied for centuries. If the exact positions of the joints of such a framework are known, the rank of the so called rigidity matrix determines whether the framework is rigid1. If the underlying graph is given only, the rigidity of the framework cannot always be determined: if certain conditions (depending on the dimension of the space) are not satisfied then the framework cannot be rigid, no matter what the actual positions of the joints are, otherwise rigidity can be realized by some (in fact, almost all) positions of the joints. These graph theoretic conditions can be checked in polynomial time for the 1- and 2-dimensional frameworks, while the complexity questions are mainly open for higher dimensions. For surveys of such results the reader is referred to [6, 13, 16].

Square grids with long “diagonals”

Bolker and Crapo gave a graph theoretical model of square grid frameworks with diagonal rods of certain squares. Baglivo and Graver solved the problem of tensegrity frameworks where diagonal cables may be used in the square grid to make it rigid. The problem of one-story buildings in both cases can be reduced to the planar problems. These results are generalized if some longer rods, respectively some longer cables are also permitted.

Elementary strong maps of graphic matroids, II

In Part I a graph theoretical tool was presented to construct elementary strong maps of graphic matroids, and it was conjectured that every such map arises in this way. An example of Jeff Kahn disproves this conjecture but leads to a generalization of the original strong map construction.

Unique solvability of linear memoryless networks-A survey

Old and new results referring to the unique solvability of networks containing independent voltage and current sources, resistors and 2-port ideal transformers and gyrators are summarized.