Evaluating the Rank Generating Function of a Graphic 2-Polymatroid

  • Authors:

  • S. D. Noble

  • Affiliations:

  • Department of Mathematical Sciences, Brunel University, Kingston Lane, Uxbridge UB8 3PH, UK (e-mail: mastsdn@brunel.ac.uk)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2006

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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.