An Introduction to Transversal Matroids
نویسنده
چکیده
1. Prefatory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Several Perspectives on Transversal Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Set systems, transversals, partial transversals, and Hall’s theorem . . . . . . . . 2 2.2. Transversal matroids via matrix encodings of set systems . . . . . . . . . . . . . . . . 3 2.3. Properties of transversal matroids that follow easily from the matrix view . 5 2.4. Background for the geometric perspective: cyclic flats . . . . . . . . . . . . . . . . . . 6 2.5. The geometric perspective on transversal matroids . . . . . . . . . . . . . . . . . . . . . . 7 3. Characterizations of Transversal Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1. Fundamental transversal matroids and the Mason-Ingleton theorem . . . . . . . 9 3.2. Sets in presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3. Multiplicities of complements of cyclic flats in presentations . . . . . . . . . . . . 12 3.4. Completion of the proof of the Mason-Ingleton theorem . . . . . . . . . . . . . . . . . 14 3.5. Several further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4. Lattice Path Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.1. Sets of lattice paths and related set systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2. Lattice path matroids; interpreting paths as bases . . . . . . . . . . . . . . . . . . . . . . . 17 4.3. Duality, direct sums, connectivity, spanning circuits, and minors . . . . . . . . . 18 5. Tutte Polynomials of Lattice Path Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.1. A brief introduction to Tutte polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2. Tutte polynomials as generating functions for basis activities . . . . . . . . . . . . 22 5.3. Basis activities in lattice path matroids; computing Tutte polynomials . . . . 23 6. Further comments and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7. Supplement: Hall’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
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