Lucchesi C. Perfect Matchings. A Theory of Matching Covered Graphs 2024

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Lucchesi C. Perfect Matchings. A Theory of Matching Covered Graphs 2024

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Textbook in PDF format Beginning with its origins in the pioneering work of W.T. Tutte in 1947, this monograph systematically traces through some of the impressive developments in matching theory. A graph is matchable if it has a perfect matching. A matching covered graph is a connected graph on at least two vertices in which each edge is covered by some perfect matching. The theory of matching covered graphs, though of relatively recent vintage, has an array of interesting results with elegant proofs, several surprising applications and challenging unsolved problems. The aim of this book is to present the material in a well-organized manner with plenty of examples and illustrations so as to make it accessible to undergraduates, and also to unify the existing theory and point out new avenues to explore so as to make it attractive to graduate students. Foreword Preface Terminology and Notation Basic Theory Perfect Matchings Matching Covered Graphs Canonical Partitions Tight Cuts Characterizations of Bricks and Braces The Perfect Matching Polytope Solid Bricks Dependence Relation and Removable Classes Dependence Classes in Bricks Removable Classes in Solid Bricks Ear Decompositions Matching Minors Brick and Brace Generation A Conjecture of Lov´asz Concerning Bricks Robust Cuts in Bricks b-Invariant Edges in Bricks The Matching Lattice and Optimal Ear Decompositions Thin Edges in Bricks and Braces Strictly Thin Edges in Bricks and Braces Pfaffian Orientations Pfaffian Orientations Similarity of Orientations and Characteristic Orientations Excluded-Minor Characterizations of Pfaffian Graphs References List of Figures Glossary Index