Let G denote the graph of the generic rigid plane skeletal system S. Let G consist of n vertices ... more Let G denote the graph of the generic rigid plane skeletal system S. Let G consist of n vertices and e edges. Of course, e? 2n-3 and we may suppose equality. (For otherwise consider 2n-3 edges whose rows form a maximal set of independent vectors in A of [8,(l)] and delete the other rows. This procedure does not affect d(S).) Laman's theorem [4] states that a graph G is minimal stiff if and only if IE(G)(= 2 j V(G)/-3 and, for every nonempty subgraph G' of G, IE(G')I 12 jV(G'>l-3. This is only a 'good characterization'. However, the following equivalent form [5, Corollary 21 gives a polynomial algorithm: (PI) Doubling any edge of G results in a graph which is the union of two trees. Consider the following, seemingly stronger, property:
This is the second volume of the procedings of the second European Congress of Mathematics. Volum... more This is the second volume of the procedings of the second European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners. Together with volume II it contains a collection of contributions by the invited lecturers. Finally, volume II also presents reports on some of the Round Table discussions. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician. Contributors: Vol. I: N. Alon, L. Ambrosio, K. Astala, R. Benedetti, Ch. Bessenrodt, F. Bethuel, P. Bjørstad, E. Bolthausen, J. Bricmont, A. Kupiainen, D. Burago, L. Caporaso, U. Dierkes, I. Dynnikov, L.H. Eliasson, W.T. Gowers, H. Hedenmalm, A. Huber, J. Kaczorowski, J. Kollár, D.O. Kramkov, A.N. Shiryaev, C. Lescop, R. März. Vol. II: J. Matousek, D. McDuff, A.S. Merkurjev, V. Milman, St. Müller, T. Nowicki, E. Olivieri, E. Scoppola, V.P. Platonov, J. Pöschel, L. Polterovich , L. Pyber, N. Simányi, J.P. Solovej, A. Stipsicz, G. Tardos, J.-P. Tignol, A.P. Veselov, E. Zuazua
Periodica polytechnica. Electrical engineering and computer science /, May 5, 2017
Consider a linear network composed of 2-terminal devices. Its interconnection structure is descri... more Consider a linear network composed of 2-terminal devices. Its interconnection structure is described by a graph G. The voltages or the currents of a subset of devices can independently be prescribed if and only if the subset of the corresponding edges in the graph G is circuit-free or cut set free, respectively. This classical result of Kirchhoff can be generalized for networks containing multiterminal devices as well: the independence structure can be described by the circuits and cut sets of a more general abstract mathematical structure, a matroid M. However, these matroids will not always be graphic. Using some recent mathematical results for characterizing graphic structures among the matroids, here we give a physical characterization of subclasses of those active networks where M happens to be graphic.
Abstract The matroid parity problem is known to be unsolvable in polynomial time in the general c... more Abstract The matroid parity problem is known to be unsolvable in polynomial time in the general case and is polynomial for linearly represented matroids. Its generalization asks if there exists an independent set in a matroid, intersecting k-tuples in a given way. The known polynomial, non-polynomial and NP-hard special cases are summarized and some open cases are mentioned.
There are plenty of NP-complete problems in very large scale integrated design, like channel rout... more There are plenty of NP-complete problems in very large scale integrated design, like channel routing or switchbox routing in the two-layer Manhattan model (2Mm, for short). However, there are quite a few polynomially solvable problems as well. Some of them (like the single row routing in 2Mm) are "classical" results; in a past survey [36] we presented some more recent ones, including: 1. a linear time channel routing algorithm in the unconstrained two-layer model; 2. a linear time switchbox routing algorithm in the unconstrained multilayer model; and 3. a linear time solution of the so called gamma routing problem in 2Mm. (This latter means that all the terminals to be interconnected are situated at two adjacent sides of a rectangular routing area, thus forming a shape. Just like channel routing, it is a special case of switchbox routing, and contains single row routing as a special case.) In the present survey talk we also mention some results from the last three years, including 4. some negative results (NP-completeness) in the multilayer Manhattan model and a channel routing algorithm if the number of layers is even; 5. an interesting relation between channel routing and multiprocessor scheduling; and 6. some improvements of 1-3 above. We present the (positive and negative) results in a systematic way, taking into account two hierarchies, namely that of geometry (what to route) and technology (how to route) at the same time.
ABSTRACT A kutatási munka során az alábbi résztémákban értünk el eredményeket: - mobil IP, - all ... more ABSTRACT A kutatási munka során az alábbi résztémákban értünk el eredményeket: - mobil IP, - all IP hálózatok, - útkeresési algoritmusok, - hívásátadási algoritmusok, - mobil technológiák együttműködése, - a szolgáltatás minősége (QoS), - a mobil és informatikai hálózatok és rendszerek biztonsági kérdései, - több-felhasználós vétel, - kódosztásos többszörös hozzáférés, - forgalmi modellezés, - kódkonstrukció kódosztásos technológiákhoz, - kvantum számítástechnikai eljárások, - gráfelmélet, - kombinatorikus optimalizálás. A fenti szakterületeken végzett kutatásaink eredményei közül azokat emeljük ki, amelyeket az alábbi témákban értünk el: - A heterogén mobil hálózatok együttműködési problémái, - A mobil Internet Protokoll alkalmazásával kapcsolatos vizsgálatok, - Többfelhasználós detekciós módszerek a kódosztásos többszörös hozzáféréses mobil rendszerekben, - A heterogén mobil hálózatok forgalmi modellezése, - A mobil informatikai és távközlési hálózatok, rendszerek és szolgáltatások - biztonsági kérdései, - Kvantum számítástechnika és mérnöki alkalmazásai, - Útkeresési és csatornakijelölési algoritmusok fejlesztése és vizsgálata mobil hálózatok számára, alkalmazott gráfelmélet. A kutatásban résztvevők az eredményeket három megvédett PhD disszertációban, egy benyújtás előtt álló akadémiai doktori értekezésben és több beadás előtt álló PhD értekezésben használták fel. A tudományos iskola publikációs listája 135 elemből áll. | The members of the Scientific School have got new results in the following scientific fields: - Mobile IP, all IP networks, - Routing algorithms, - Hand-over algorithms, - Interworking of heterogeneous mobile technologies, - Quality of services (QoS), - Security problems of mobile and information networks and systems, - Multi-user detection, - Code division multiple access, - Traffic modeling, - Code construction for code division technologies, - Quantum computing, - Graph theory, - Combinatorial optimization. On the above mentioned scientific field we have the most important results in the following areas: - Interoperability issues of heterogeneous mobile networks, - Investigations on the applicability of mobile Internet Protocol, - Multi-user detection methods in code division multiple access systems, - Traffic models of heterogeneous mobile networks, - Security issues of mobile information and telecommunication networks, systems and services, - Quantum computing and its engineering applications, - Development and research of routing and channel assigning algorithms for mobile networks, application of the graph theory. The participants of the research used their results in three defended PhD theses, in a dissertation for DSc title, and in some other PhD theses before the final process. The number of the publications of the Scientific School is 135.
In this paper we introduce a flexible software framework for the easy implementation and evaluati... more In this paper we introduce a flexible software framework for the easy implementation and evaluation of VLSI routing algorithms and the visualization of routing results. Moreover, we present some new results in the theory of VLSI routing algorithms for rectangular grids, as well as the reformulation of some classical algorithms for triangular grids.
Periodica Polytechnica Electrical Engineering and Computer Science, 2021
The genericity assumption, supposing that the nonzero parameters of a system are algebraically in... more The genericity assumption, supposing that the nonzero parameters of a system are algebraically independent transcendentals over the field of the rationals, often helps for the mathematical modelling of linear systems. Without this condition nonzero expansion members of a determinant can cancel out each other, decreasing the rank of a matrix. In this note we show that under some circumstances an increase is also possible. This counterintuitive phenomenon is explained using some tools from matroid theory, and is illustrated by a classical network of Carlin and Youla.
Több mint fél évszázada vetődött fel a probléma, hogyan lehet eldönteni egy n-kapuról, hogy van-e... more Több mint fél évszázada vetődött fel a probléma, hogyan lehet eldönteni egy n-kapuról, hogy van-e hibrid leírása. A problémát Iri és Tomizawa matroidelméleti eszközökkel még a hetvenes években megoldotta, itt ennek egy természetes általánosítását vizsgáljuk. Megmutatjuk, hogy az általánosabb kérdéshez további -erősebb -matroidelméleti eszközök kellenek, végül megvizsgáljuk, hogy a rúdszerkezetek és a villamos hálózatok közötti régen ismert analógia ebben az esetben milyen statikai kérdéshez vezet.
Tensegrity frameworks are defined on a set of points in Rd and consist of bars, cables, and strut... more Tensegrity 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 labelled 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. We show that a graph can be labelled as a rigid tensegrity graph in Rd containing only cables and struts if and only if it is redundantly rigid in Rd. When d = 2 we give an efficient combinatorial algorithm for finding a rigid cablestrut labelling. We also obtain some partial results on the characterization of rigid tensegrity graphs in R2.
There is a conjecture that if the union (also called sum) of graphic matroids is not graphic then... more There is a conjecture that if the union (also called sum) of graphic matroids is not graphic then it is nonbinary. Some special cases have been proved only, for example if several copies of the same graphic matroid are given. If there are two matroids and the first one can either be represented by a graph with two points, or is the direct sum of a circuit and some loops, then a necessary and sufficient condition is known for the other matroid to ensure the graphicity of the union and the above conjecture holds for these cases. We have proved the sufficiency of this condition for the graphicity of the union of two arbitrary graphic matroids. Then we present a weaker necessary condition which is of similar character. Finally we suggest a more general framework of the study of such questions by introducing matroid classes formed by those graphic (or arbitrary) matroids whose union with any graphic (or arbitrary) matroid is graphic (or either graphic or nonbinary).
Let G denote the graph of the generic rigid plane skeletal system S. Let G consist of n vertices ... more Let G denote the graph of the generic rigid plane skeletal system S. Let G consist of n vertices and e edges. Of course, e? 2n-3 and we may suppose equality. (For otherwise consider 2n-3 edges whose rows form a maximal set of independent vectors in A of [8,(l)] and delete the other rows. This procedure does not affect d(S).) Laman's theorem [4] states that a graph G is minimal stiff if and only if IE(G)(= 2 j V(G)/-3 and, for every nonempty subgraph G' of G, IE(G')I 12 jV(G'>l-3. This is only a 'good characterization'. However, the following equivalent form [5, Corollary 21 gives a polynomial algorithm: (PI) Doubling any edge of G results in a graph which is the union of two trees. Consider the following, seemingly stronger, property:
This is the second volume of the procedings of the second European Congress of Mathematics. Volum... more This is the second volume of the procedings of the second European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners. Together with volume II it contains a collection of contributions by the invited lecturers. Finally, volume II also presents reports on some of the Round Table discussions. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician. Contributors: Vol. I: N. Alon, L. Ambrosio, K. Astala, R. Benedetti, Ch. Bessenrodt, F. Bethuel, P. Bjørstad, E. Bolthausen, J. Bricmont, A. Kupiainen, D. Burago, L. Caporaso, U. Dierkes, I. Dynnikov, L.H. Eliasson, W.T. Gowers, H. Hedenmalm, A. Huber, J. Kaczorowski, J. Kollár, D.O. Kramkov, A.N. Shiryaev, C. Lescop, R. März. Vol. II: J. Matousek, D. McDuff, A.S. Merkurjev, V. Milman, St. Müller, T. Nowicki, E. Olivieri, E. Scoppola, V.P. Platonov, J. Pöschel, L. Polterovich , L. Pyber, N. Simányi, J.P. Solovej, A. Stipsicz, G. Tardos, J.-P. Tignol, A.P. Veselov, E. Zuazua
Periodica polytechnica. Electrical engineering and computer science /, May 5, 2017
Consider a linear network composed of 2-terminal devices. Its interconnection structure is descri... more Consider a linear network composed of 2-terminal devices. Its interconnection structure is described by a graph G. The voltages or the currents of a subset of devices can independently be prescribed if and only if the subset of the corresponding edges in the graph G is circuit-free or cut set free, respectively. This classical result of Kirchhoff can be generalized for networks containing multiterminal devices as well: the independence structure can be described by the circuits and cut sets of a more general abstract mathematical structure, a matroid M. However, these matroids will not always be graphic. Using some recent mathematical results for characterizing graphic structures among the matroids, here we give a physical characterization of subclasses of those active networks where M happens to be graphic.
Abstract The matroid parity problem is known to be unsolvable in polynomial time in the general c... more Abstract The matroid parity problem is known to be unsolvable in polynomial time in the general case and is polynomial for linearly represented matroids. Its generalization asks if there exists an independent set in a matroid, intersecting k-tuples in a given way. The known polynomial, non-polynomial and NP-hard special cases are summarized and some open cases are mentioned.
There are plenty of NP-complete problems in very large scale integrated design, like channel rout... more There are plenty of NP-complete problems in very large scale integrated design, like channel routing or switchbox routing in the two-layer Manhattan model (2Mm, for short). However, there are quite a few polynomially solvable problems as well. Some of them (like the single row routing in 2Mm) are "classical" results; in a past survey [36] we presented some more recent ones, including: 1. a linear time channel routing algorithm in the unconstrained two-layer model; 2. a linear time switchbox routing algorithm in the unconstrained multilayer model; and 3. a linear time solution of the so called gamma routing problem in 2Mm. (This latter means that all the terminals to be interconnected are situated at two adjacent sides of a rectangular routing area, thus forming a shape. Just like channel routing, it is a special case of switchbox routing, and contains single row routing as a special case.) In the present survey talk we also mention some results from the last three years, including 4. some negative results (NP-completeness) in the multilayer Manhattan model and a channel routing algorithm if the number of layers is even; 5. an interesting relation between channel routing and multiprocessor scheduling; and 6. some improvements of 1-3 above. We present the (positive and negative) results in a systematic way, taking into account two hierarchies, namely that of geometry (what to route) and technology (how to route) at the same time.
ABSTRACT A kutatási munka során az alábbi résztémákban értünk el eredményeket: - mobil IP, - all ... more ABSTRACT A kutatási munka során az alábbi résztémákban értünk el eredményeket: - mobil IP, - all IP hálózatok, - útkeresési algoritmusok, - hívásátadási algoritmusok, - mobil technológiák együttműködése, - a szolgáltatás minősége (QoS), - a mobil és informatikai hálózatok és rendszerek biztonsági kérdései, - több-felhasználós vétel, - kódosztásos többszörös hozzáférés, - forgalmi modellezés, - kódkonstrukció kódosztásos technológiákhoz, - kvantum számítástechnikai eljárások, - gráfelmélet, - kombinatorikus optimalizálás. A fenti szakterületeken végzett kutatásaink eredményei közül azokat emeljük ki, amelyeket az alábbi témákban értünk el: - A heterogén mobil hálózatok együttműködési problémái, - A mobil Internet Protokoll alkalmazásával kapcsolatos vizsgálatok, - Többfelhasználós detekciós módszerek a kódosztásos többszörös hozzáféréses mobil rendszerekben, - A heterogén mobil hálózatok forgalmi modellezése, - A mobil informatikai és távközlési hálózatok, rendszerek és szolgáltatások - biztonsági kérdései, - Kvantum számítástechnika és mérnöki alkalmazásai, - Útkeresési és csatornakijelölési algoritmusok fejlesztése és vizsgálata mobil hálózatok számára, alkalmazott gráfelmélet. A kutatásban résztvevők az eredményeket három megvédett PhD disszertációban, egy benyújtás előtt álló akadémiai doktori értekezésben és több beadás előtt álló PhD értekezésben használták fel. A tudományos iskola publikációs listája 135 elemből áll. | The members of the Scientific School have got new results in the following scientific fields: - Mobile IP, all IP networks, - Routing algorithms, - Hand-over algorithms, - Interworking of heterogeneous mobile technologies, - Quality of services (QoS), - Security problems of mobile and information networks and systems, - Multi-user detection, - Code division multiple access, - Traffic modeling, - Code construction for code division technologies, - Quantum computing, - Graph theory, - Combinatorial optimization. On the above mentioned scientific field we have the most important results in the following areas: - Interoperability issues of heterogeneous mobile networks, - Investigations on the applicability of mobile Internet Protocol, - Multi-user detection methods in code division multiple access systems, - Traffic models of heterogeneous mobile networks, - Security issues of mobile information and telecommunication networks, systems and services, - Quantum computing and its engineering applications, - Development and research of routing and channel assigning algorithms for mobile networks, application of the graph theory. The participants of the research used their results in three defended PhD theses, in a dissertation for DSc title, and in some other PhD theses before the final process. The number of the publications of the Scientific School is 135.
In this paper we introduce a flexible software framework for the easy implementation and evaluati... more In this paper we introduce a flexible software framework for the easy implementation and evaluation of VLSI routing algorithms and the visualization of routing results. Moreover, we present some new results in the theory of VLSI routing algorithms for rectangular grids, as well as the reformulation of some classical algorithms for triangular grids.
Periodica Polytechnica Electrical Engineering and Computer Science, 2021
The genericity assumption, supposing that the nonzero parameters of a system are algebraically in... more The genericity assumption, supposing that the nonzero parameters of a system are algebraically independent transcendentals over the field of the rationals, often helps for the mathematical modelling of linear systems. Without this condition nonzero expansion members of a determinant can cancel out each other, decreasing the rank of a matrix. In this note we show that under some circumstances an increase is also possible. This counterintuitive phenomenon is explained using some tools from matroid theory, and is illustrated by a classical network of Carlin and Youla.
Több mint fél évszázada vetődött fel a probléma, hogyan lehet eldönteni egy n-kapuról, hogy van-e... more Több mint fél évszázada vetődött fel a probléma, hogyan lehet eldönteni egy n-kapuról, hogy van-e hibrid leírása. A problémát Iri és Tomizawa matroidelméleti eszközökkel még a hetvenes években megoldotta, itt ennek egy természetes általánosítását vizsgáljuk. Megmutatjuk, hogy az általánosabb kérdéshez további -erősebb -matroidelméleti eszközök kellenek, végül megvizsgáljuk, hogy a rúdszerkezetek és a villamos hálózatok közötti régen ismert analógia ebben az esetben milyen statikai kérdéshez vezet.
Tensegrity frameworks are defined on a set of points in Rd and consist of bars, cables, and strut... more Tensegrity 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 labelled 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. We show that a graph can be labelled as a rigid tensegrity graph in Rd containing only cables and struts if and only if it is redundantly rigid in Rd. When d = 2 we give an efficient combinatorial algorithm for finding a rigid cablestrut labelling. We also obtain some partial results on the characterization of rigid tensegrity graphs in R2.
There is a conjecture that if the union (also called sum) of graphic matroids is not graphic then... more There is a conjecture that if the union (also called sum) of graphic matroids is not graphic then it is nonbinary. Some special cases have been proved only, for example if several copies of the same graphic matroid are given. If there are two matroids and the first one can either be represented by a graph with two points, or is the direct sum of a circuit and some loops, then a necessary and sufficient condition is known for the other matroid to ensure the graphicity of the union and the above conjecture holds for these cases. We have proved the sufficiency of this condition for the graphicity of the union of two arbitrary graphic matroids. Then we present a weaker necessary condition which is of similar character. Finally we suggest a more general framework of the study of such questions by introducing matroid classes formed by those graphic (or arbitrary) matroids whose union with any graphic (or arbitrary) matroid is graphic (or either graphic or nonbinary).
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