Matroid Theory"The theory of matroids connects disparate branches of combinatorial theory and algebra such as graph and lattice theory, combinatorial optimization, and linear algebra. Aimed at advanced undergraduate and graduate students, this text is one of the earliest substantial works on matroid theory. Its author, D. J. A. Welsh, Professor of Mathematics at Oxford University, has exercised a profound influence over the theory's development. The first half of the text describes standard examples and investigation results, using elementary proofs to develop basic matroid properties and referring readers to the literature for more complex proofs. The second half advances to a more sophisticated treatment, addressing a variety of research topics. Praised by the Bulletin of the American Mathematical Society as 'a useful resource for both the novice and the expert', this text features numerous helpful exercises."--Publisher's description. |
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A₁ abstract dual algebraic axioms B₁ base orderable binary matroid bipartite graph blocks C₁ C₂ cardinality Chapter chromatic polynomial circuit cocircuit cocycle cographic Combinatorial Theory contains COROLLARY Crapo cycle matroid D₁ defined denote digraph disjoint e₁ edges of G elements Example EXERCISES exists F₁ field F finite G₁ geometric lattice graph G graphic greedy algorithm H₁ Hence homeomorphic hyperplanes independence space independent sets Ingleton integer intersection isomorphic L₁ lemma Let G linear loop M₁ M₂ matrix matroid design matroid of rank maximal Möbius function modular modular lattice P₁ partition planar graph polymatroid problem projective space Proof of Theorem Prove Rado's theorem rank function representable result S₁ satisfies set of edges simple matroid spanning Steiner system strict gammoid strong map subgraph submodular subspace Suppose T₁ transversal matroid Tutte Tutte polynomial u₁ uniform matroid vector space vertex set vertices x₁