Evaluating the Rank Generating Function of a Graphic 2-Polymatroid

Abstract

We consider the complexity of the two-variable rank generating function, $S$, of a graphic 2-polymatroid. For a graph $G$, $S$ is the generating function for the number of subsets of edges of $G$ having a particular size and incident with a particular number of vertices of $G$. We show that for any $x,y \in \mathbb{Q}$ with $xy \not = 1$, it is $\#$P-hard to evaluate $S$ at $(x,y)$. We also consider the $k$-thickening of a graph and computing $S$ for the $k$-thickening of a graph.