The Tutte polynomial of a ported matroid

Las Vergnas’ generalizations of the Tutte polynomial are studied as follows. The theory of Tutte-Grothendieck matroid invariantsfis modified so the Tutte decomposition ,f(M) =f(M\e) +f( M/e) is applied only when e $ P (and e is neither a loop nor an isthmus) where P is a distinguished set of points called ports. The resulting “P-ported” Tutte polynomial tp has variables I, w; q,, qZ, . . . . qm; the q’s 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. '8 1989 Academic Press, Inc.