The Tutte polynomial of a ported matroid

are connected matroids on subsets of P. We express tp of the P-ported matroid sum and cosum of M, and M,, only taken when M, n M, c_ P, in terms of rP(M1) and r,(M,). The behavior of rp(M) under contraction/deletion of points in P is given. Our development is based on a P-ported rank generating function. The major results are generalizations of work of Brylawski on series and parallel connections. Relationships to the geometric lattice of M, to Las Vergnas’ Tutte polynomial of a matroid pointed by a family of sets, and to electrical network theory are given.

[1]  R. Bixby,et al.  A composition for matroids , 1975 .

[2]  András Recski,et al.  On the sum of matroids, III , 1981, Discret. Math..

[3]  Paul D. Seymour,et al.  Decomposition of regular matroids , 1980, J. Comb. Theory, Ser. B.

[4]  M. Iri,et al.  Applications of Matroid Theory , 1982, ISMP.

[5]  H. Crapo,et al.  The Tutte polynomial , 1969, 1707.03459.

[6]  Joseph P. S. Kung,et al.  Bimatroids and invariants , 1978 .

[7]  W. Cunningham.  Polyhedra for Composed Independence Systems , 1982 .

[8]  C. Satyanarayana,et al.  Applied graph theory: Graphs and electrical networks , 1976, Proceedings of the IEEE.

[9]  Tom Brylawski,et al.  A decomposition for combinatorial geometries , 1972 .

[10]  Alexander Schrijver,et al.  Matroids and linking systems , 1979, J. Comb. Theory B.

[11]  S. Chaiken.  A Combinatorial Proof of the All Minors Matrix Tree Theorem , 1982 .

[12]  R. Duffin.  Topology of series-parallel networks , 1965 .

[13]  Tom Brylawski,et al.  Modular constructions for combinatorial geometries , 1975 .

[14]  J. Edmonds,et al.  A Combinatorial Decomposition Theory , 1980, Canadian Journal of Mathematics.

[15]  Matrix Operations Induced by Network Connections , 1975 .

[16]  Michel Las Vergnas,et al.  Acyclic and totally cyclic orientations of combinatorial geometries , 1977, Discret. Math..

[17]  Tom Brylawski,et al.  A combinatorial model for series-parallel networks , 1971 .

[18]  G. Dantzig,et al.  Mathematics of the decision sciences , 1968 .

[19]  Michel Las Vergnas,et al.  On the Tutte Polynomial of a Morphism of Matroids , 1980 .

[20]  W. H. Cunningham,et al.  Chords and disjoint paths in matroids , 1977, Discret. Math..

[21]  László Lovász,et al.  On the sum of matroids , 1973 .

[22]  G. Rota.  On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[23]  Richard P. Stanley,et al.  Modular elements of geometric lattices , 1971 .