zbMATH — the first resource for mathematics

Matroid theory. 2nd ed. (English) Zbl 1254.05002
Oxford Graduate Texts in Mathematics 21. Oxford: Oxford University Press (ISBN 978-0-19-856694-6/hbk; 978-0-19-960339-8/pbk). xiii, 684 p. (2011).
This book is a major revision of the first edition reviewed in [Zbl 0784.05002], which appeared in 1992, see also [Zbl 1115.05001] for the paperback edition which appeared in 2006. Chapters 1-7 are close to their original form but have many minor revisions and some interesting additions, as, for example, the new section on Dowling matroids (section 6.10), or the new section 7.4 entitled ”A non-commutative operation” describing how to stick together two matroids in a non-commutative fashion. Chapter 8, which deals with higher connectivity, has been significantly revised and lengthened with the inclusion of Tutte’s Linking Theorem (Section 8.5), which is a matroid analogue of Menger’s Theorem. Also added to that chapter is a proof of Cunningham and Edmond’s tree decomposition of matroids that are 2-connected but not 3-connected. Chapter 9, on binary matroids, includes now an expanded discussion on the 3-sum operation. Chapter 10 of this edition essentially combines and revises Chapters 10 and 13 of the first edition. It now includes proofs of the excluded-minor theorems for regular, ternary, and graphic matroids. The order of Chapters 11 (Submodular functions and matroid union) and 12 (The Splitter Theorem) is reversed from that in the first edition, and Section 11.5 (Generalizations of delta-wye exchange) is new. The final chapter on unsolved problems is updated from the first edition and includes several new problems. Through the author’s website [https://www.math.lsu.edu/~oxley/] one can access a file containing an errata and update on conjectures, problems, and references from the second edition of the book in pdf format.

05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05B35 Combinatorial aspects of matroids and geometric lattices