Abstract
There is an extensive body of literature on the subject of transversal matroids (with the partial transversals of a family of sets as independent sets) and the more general theory of matroids induced from graphs. This paper reviews the known results and some still unsolved problems. The topics discussed include: characterizations of transversal matroids and of their presentations, standard operations applied to transversal matroids, gammoids and strict gammoids, base-orderability of matroids. The notion of a complete class of matroids (the gammoids being the simplest such class) is introduced and discussed.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J. A. Bondy, Transversal matroids, base-orderable matroids and graphs, Quart.J.Math.(Oxford)(2) 23 (1972), 81–89.
______, Presentations of transversal matroids, J. London Math.Soc. (2) 5 (1972), 289–92.
______, and D.J.A. Welsh, Some results on transversal matroids and constructions of identically self-dual matroids, Quart.J.Math. (Oxford)(2) 22 (1971), 435–51.
R. A. Brualdi, Comments on bases in dependence structures, Bull.Australian Math.Soc. 7 (1969), 161–67.
______, Induced matroids, Proc.American Math.Soc. 29 (1971), 221–31.
______, Fundamental transversal matroids, Proc. American Math. Soc. 45 (1974) 151–56.
______, Matroids induced by directed graphs, a survey, Proc. 2nd Czech.Graph Theory Symp., Prague (1974), 115–34.
______, and G.W. Dinolt, Characterizations of transversal matroids and their presentations, J.Comb.Theory 12 (1972), 268–86.
______ _______, Truncations of principal geometries, Discrete Math. 12 (1975), 113–38.
______, and J.H. Mason, Transversal matroids and Hall’s theorem, Pacific J.Math, 41 (1972), 601–13.
______, and E.B. Scrimger, Exchange systems, matchings and transversals, J.Comb.Theory 5 (1968), 244–57.
T.H. Brylawski, A combinatorial model for series-parallel networks, Trans.American Math.Soc. 154 (1971), 1–22.
______, An affine representation for transversal geometries, Studies in Appl.Math. 54 (1975), 143–60.
H.H. Crapo and G.-C. Rota, Combinatorial Geometries (M.I.T. Press, 1970 ).
J. Davies, Some problems in matroid theory, D.Phil.thesis, Univ. of Oxford (1975).
J. de Sousa and D.J.A. Welsh, A characterization of binary- transversal structures, J.Math.Anal.Appl. 40 (1972), 55–59.
T.A. Dowling and D.G. Kelly, Elementary strong maps and transversal geometries, Discrete Math. 7. (1974), 209–24.
J.R. Edmonds and D.R. Fulkerson, Transversals and matroid partition, J. Res. Nat. Bur. Standards 69B (1965), 147–53.
A.W. Ingleton, Conditions for representability and transversality of matroids, in Théorie des Matroides, ed.C.P.Bruter (Springer, 1971 ), 62–66.
______, A geometrical characterization of transversal independence structures, Bull.London Math.Soc. 3 (1971), 47–51.
______, Non-base-orderable matroids, Proc. 5th British Combinatorial Conf., Aberdeen (1975), 355–60.
_______, and M.J. Piff, Gammoids and transversal matroids, J.Comb.Theory 15 (1973), 51–68.
M. Las Vergnas, Matroides orientables, preprint.
J.H. Mason, A characterization of transversal independence spaces, in Théorie des Matroides, ed.C.P.Bruter (Springer, 1971 ), 86–94.
______, On a class of matroids arising from paths in graphs, Proc.London Math.Soc. (3) 25 (1972), 55–74.
L.R. Matthews, Bicircular matroids, preprint.
C. McDiarmid, Strict gammoids and rank functions, Bull.London Math.Soc. 4 (1972), 196–98.
L. Mirsky, Transversal Theory (Academic Press, 1971).
______, and H. Perfect, Applications of the notion of linear independence to problems in combinatorial analysis, J.Comb. Theory 2 (1967), 327–57.
H. Narayanan and M.N. Vartak, Gammoids, base-orderable matroids and series-parallel networks, preprint.
H. Perfect, Applications of Menger’s graph theorem, J.Math. Anal.Appl. 22 (1968), 96–110.
______, Independence spaces and combinatorial problems, Proc.London Math.Soc. (3) 19 (1969), 17–30.
M.J. Piff and D.J.A. Welsh, On the vector representation of matroids, J.London Math.Soc. (2) 2 (1970), 284–88.
J.S. Pym, The linking of sets in graphs, J.London Math.Soc. 44 (1969), 542–50.
R. Rado, A theorem on independence relations, Quart.J.Math. (Oxford) 13 (1942), 83–89.
A. Recski, On partitional matroids with applications, Proc. Colloq.Math.Soc.János Bolyai, Keszthely (1973), 1169–79.
J.M.S. Simões-Pereira, On subgraphs as matroid cells, Math.Z. 127 (1972), 315–22.
D.R. Woodall, Linking in graphoids, preprint.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1977 D. Reidel Publishing Company, Dordrecht-Holland
About this paper
Cite this paper
Ingleton, A.W. (1977). Transversal Matroids and Related Structures. In: Aigner, M. (eds) Higher Combinatorics. NATO Advanced Study Institutes Series, vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1220-1_8
Download citation
DOI: https://doi.org/10.1007/978-94-010-1220-1_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-1222-5
Online ISBN: 978-94-010-1220-1
eBook Packages: Springer Book Archive