A recursive definition for the polymatroid Tutte polynomial
Xiaxia Guan, Xian'an Jin, Weiling Yang
TL;DR
This work establishes that the polymatroid Tutte polynomial $\mathscr{T}_{P}(x,y)$, previously defined via bases, rank generating functions, and a deletion-contraction framework, can equivalently be defined by a recursive formula $\mathscr{T}'_{P}(x,y)$ that uses deletions, contractions, and projections of polymatroids. The authors prove well-definedness of this recursion by induction on the ground-set size and a comprehensive case analysis, ensuring the resulting polynomial is independent of processing order. They show $\mathscr{T}'_{P}(x,y)$ coincides with the established $\mathscr{T}_{P}(x,y)$, thereby providing a self-contained, order-invariant recursive definition for polymatroid Tutte polynomials that generalizes matroid results. This solidifies the theoretical robustness of the polymatroid Tutte invariant and offers a direct recursive construction for computational purposes. The work deepens the connection between polymatroid theory and Tutte-like invariants and reinforces the unifying role of deletion-contraction techniques.
Abstract
The Tutte polynomial is a significant invariant of graphs and matroids. It is well-known that it has three equivalent definitions: bases expansion, rank generating function, and deletion-contraction formula. The polymatroid Tutte polynomial $\mathscr{T}_{P}$ generalizes the Tutte polynomial from matroids to polymatroids $P$. In \emph{[Adv. Math. 402 (2022) 108355.]} and \emph{[J. Combin. Theory Ser. A 188 (2022) 105584]}, the authors provided bases expansion and rank generating function constructions for $\mathscr{T}_{P}$, respectively. In \emph{[Int. Math. Res. Not. 19 (2025) rnaf302]}, a recursive formula for $\mathscr{T}_{P}$ was obtained. In this paper, we show that the recursive formula itself can be used to define the polymatroid Tutte polynomial independently.