The Tutte polynomial of a ported matroid
Las Vergnas' generalizations of the Tutte polynomial are studied as follows. The theory of Tutte-Grothendieck matroid invariants f is modified so the Tutte decomposition \(f(M)=f(M\setminus e)+f(M/e)\) is applied only when \(e\not\in P\) (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 \(t_ P\) has variables \(z,w;q_ 1,q_ 2,...,q_ m\); the q's are connected matroids on subsets of P. We express \(t_ P\) of the P-ported matroid sum and cosum of \(M_ 1\) and \(M_ 2\), only taken when \(M_ 1\cap M_ 2\subseteq P\), in terms of \(t_ P(M_ 1)\) and \(t_ P(M_ 2)\). The behavior of \(t_ P(M)\) 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.
scientific article; zbMATH DE number 3993603
scientific article