Planar graphs and matroids

Weinberg, Louis

doi:10.1007/bfb0067384

Graph Theory and Applications

1972

DOI: 10.1007/bfb0067384

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Planar graphs and matroids

Louis Weinberg

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1976

1981

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Cited by 3 publications

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Kuratowski's theorem

Thomassen

1981

Journal of Graph Theory

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We present three short proofs of Kuratowski's theorem on planarity of graphs and discuss applications, extensions, and some related problems. INTRODUCTIONPlanar graphs are of great importance in graph theory. They are interesting in their own right and their chromatic, enumerative, hamiltonian, and other properties have been studied in great detail. Furthermore, planar graphs are of some importance in the study of convex polytopes since, by Steinitz's theorem, the 1-skeletons of the 3-dimensional polytopes are precisely the 3-connected planar graphs. Finally, planar graphs provide an important link between graphs and matroids.In The importance of Kuratowski's theorem is not so much its applications to planar graph theory (in fact, it is used only in relatively few results on planar graphs). It rather lies in the fact that a characterization of planar graphs in terms of essentially a finite number of forbidden subgraphs exists at all. Also, Kuratowski's theorem has a special ranking among the known planarity criteria not only because of its beauty and simplicity, but also because it implies rather easily the planarity criteria by MacLane [27] ([40]). It should also be noted that, unlike most other planarity criteria, Kuratowski's theorem provides a useful characterization of the nonplanar graphs. Finally, it seems that almost all known proofs of Kuratowski's theorem can be turned into polynomially

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Kuratowski's theorem

Thomassen

1981

Journal of Graph Theory

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Generalized networks: Networks embedded on a matroid, part I

Bruno

Weinberg

1976

Networks

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The d e f i n i t i o n of ordinary e l e c t r i c networks i s based on. graphs. Since such a formulation is inadequate i n many respects, we define a n m network concept, namely, a g e n m d i z e d MGbhkIhk., which i s a network based on a matroid. To make the paper s e l fcontained an introduction t o basic matroid theory i s given i n the f i r s t part of the paper. We then define a generalized network and formuZate the network anaZysis and network synthesis problems f o r generalized networks. A number of new r e s u l t s are obtained f o r both anaZysis and synthesis and some old r e s u l t s f o r networks on graphs are generalized t o networks on matroids. I t i s shown t h a t the principle of d u a l i t y , which does not hold f o r networks on graphs but i s v a l i d f o r generalized networks, adds power and i n s i g h t The synthesis ppoblem t h a t i s treated i s t h e crucial

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Generalized networks: Networks embedded on a matroid, part II

Bruno

Weinberg

1976

Networks

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This p a p e r r e p r e s e n t s t h e second p a r t of t h e p a p e r w i t h t h e same t i t l e which appeared i n Networks, Volume 6 , Number 1.I n t h i s p a r t w e apply t h e t h e o r y p r e s e n t e d i n P a r t I t o d e f i n e a g e n e r a l i z e d network and d e r i v e i t s i m p o r t a n t p r o p e r t i e s . RESISTANCE NETWORKS AND GENERALIZED NETWORKSI n t h i s s e c t i o n we i n t r o d u c e t h e concept of a g e n e r a l i z e d r e s i s t a n c e network and s t u d y some of i t s p r o p e r t i e s . The gene r a l i z e d network i s an e x t e n s i o n of t h e concepts of o r d i n a r y p-port r e s i s t a n c e and RLC networks t o m a t r o i d s . i n P a r t I , though w e c a r r y o u t t h e a n a l y s i s i n terms of resist a n c e networks, t h e a p p l i c a t i o n t o RLC networks i s v a l i d and given immediately by simple s u b s t i t u t i o n . A s d i s c u s s e d Resistance NetworksI n t h i s s u b s e c t i o n we d e s c r i b e a p -p o r t r e s i s t a n c e network as a p r e l i m i n a r y t o t h e d i s c u s s i o n o f t h e g e n e r a l i z e d r e s i s t a n c e network i n t h e n e x t s u b s e c t i o n . A p -p o r t r e s i s t a n c e network i s an i n t e r c o n n e c t i o n of two t y p e s of elements: p o r t elements and r e s i s t a n c e e l e m e n t s . A p o r t element i s denoted by a d i r e c t e d edge ( F i g u r e 11) and t h e convention used i s t h a t t h e d i r e c t i o n of p o s i t i v e c u r r e n c (i) c o i n c i d e s w i t h t h e d i r e c t i o n of t h e arrow. P o s i t i v e p o t e n t i a l d i f f e r e n c e ( v ) means t h a t t h e arrow p o i n t s from t h e v e r t e x of high p o t e n t i a l t o t h e v e r t e x of low p o t e n t i a l .Note t h a t t h e p r o d u c t v i r e p r e s e n t s t h e i n s t a n t aneous power d e l i v e r e d t o t h e p o r t element. A p o r t edge may t h u s b e c o n s i d e r e d as a distinguished edge. i d e n t i c a l t o a p o r t element w i t h t h e a d d i t i o n a l requirement t h a t v = i z , where 0 < z < a and z i s c a l l e d t h e resistance of t h e element.

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Planar graphs and matroids

Cited by 3 publications

References 5 publications

Kuratowski's theorem

Kuratowski's theorem

Generalized networks: Networks embedded on a matroid, part I

Generalized networks: Networks embedded on a matroid, part II

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