Binary supersolvable matroids and modular constructions

Ziegler, Günter M.

doi:10.1090/S0002-9939-1991-1068134-3

Binary supersolvable matroids and modular constructions
HTML articles powered by AMS MathViewer

by Günter M. Ziegler PDF

Proc. Amer. Math. Soc. 113 (1991), 817-829 Request permission

Abstract:

Let $\mathcal {M}$ be the class of binary matroids without a Fano plane as a submatroid. We show that every supersolvable matroid in $\mathcal {M}$ is graphic, corresponding to a chordal graph. Then we characterize the case that the modular join of two matroids is supersolvable. This is used to study modular flats and modular joins of binary supersolvable matroids. We decompose supersolvable matroids in $\mathcal {M}$ as modular joins with respect to hyperplanes. For such matroids every modular flat is contained in a maximal chain of modular flats, and thus modular joins are again supersolvable.

References

Kenneth Baclawski and Neil L. White, Higher order independence in matroids, J. London Math. Soc. (2) 19 (1979), no. 2, 193–202. MR 533317, DOI 10.1112/jlms/s2-19.2.193
F. Barahona and M. Grötschel, On the cycle polytope of a binary matroid, J. Combin. Theory Ser. B 40 (1986), no. 1, 40–62. MR 830592, DOI 10.1016/0095-8956(86)90063-8
Anders Björner and Günter M. Ziegler, Broken circuit complexes: factorizations and generalizations, J. Combin. Theory Ser. B 51 (1991), no. 1, 96–126. MR 1088629, DOI 10.1016/0095-8956(91)90008-8
R. C. Bose and R. C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes, J. Combinatorial Theory 1 (1966), 96–104. MR 197215
Tom Brylawski, Modular constructions for combinatorial geometries, Trans. Amer. Math. Soc. 203 (1975), 1–44. MR 357163, DOI 10.1090/S0002-9947-1975-0357163-6
Tom Brylawski, The broken-circuit complex, Trans. Amer. Math. Soc. 234 (1977), no. 2, 417–433. MR 468931, DOI 10.1090/S0002-9947-1977-0468931-6
Reinhard Diestel, Simplicial decompositions, tree-decompositions and graph minors, Ars Combin. 25 (1988), no. C, 97–104. Eleventh British Combinatorial Conference (London, 1987). MR 943381
G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71–76. MR 130190, DOI 10.1007/BF02992776
Martin Charles Golumbic, Algorithmic graph theory and perfect graphs, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. With a foreword by Claude Berge. MR 562306
Mark D. Halsey, Line-closed combinatorial geometries, Discrete Math. 65 (1987), no. 3, 245–248. MR 897649, DOI 10.1016/0012-365X(87)90056-2
I. Heller, On linear systems with integral valued solutions, Pacific J. Math. 7 (1957), 1351–1364. MR 94381
Joseph P. S. Kung, Growth rates and critical exponents of classes of binary combinatorial geometries, Trans. Amer. Math. Soc. 293 (1986), no. 2, 837–859. MR 816330, DOI 10.1090/S0002-9947-1986-0816330-2
Bernt Lindström, On strong joins and pushout of combinatorial geometries, J. Combin. Theory Ser. A 25 (1978), no. 1, 77–79. MR 491268, DOI 10.1016/0097-3165(78)90035-3
U. S. R. Murty, Extremal matroids with forbidden restrictions and minors (synopsis), Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), Congressus Numerantium, No. XVII, Utilitas Math., Winnipeg, Man., 1976, pp. 463–468. MR 0444500
P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), no. 3, 305–359. MR 579077, DOI 10.1016/0095-8956(80)90075-1
Richard P. Stanley, Modular elements of geometric lattices, Algebra Universalis 1 (1971/72), 214–217. MR 295976, DOI 10.1007/BF02944981
R. P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197–217. MR 309815, DOI 10.1007/BF02945028
D. J. A. Welsh, Matroid theory, L. M. S. Monographs, No. 8, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0427112

Theory of matroids

Combinatorial geometries