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arXiv · 1605.01455

Connectivity Functions and Polymatroids

Susan Jowett·Songbao Mo·Geoff Whittle
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Abstract

A {\em connectivity function on} a set $E$ is a function $\lambda:2^E\rightarrow \mathbb R$ such that $\lambda(\emptyset)=0$, that $\lambda(X)=\lambda(E-X)$ for all $X\subseteq E$ and that $\lambda(X\cap Y)+\lambda(X\cup Y)\leq \lambda(X)+\lambda(Y)$ for all $X,Y \subseteq E$. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. We introduce a notion of duality for polymatroids and prove that every connectivity function is the connectivity function of a self-dual polymatroid. We also prove that every integral connectivity function is the connectivity function of a half-integral self-dual polymatroid.

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Susan Jowett, Songbao Mo, Geoff Whittle. 2016-05-04. Connectivity Functions and Polymatroids. https://arxiv.org/abs/1605.01455

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