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It is known that that can be several trees in a graph that follow the conditions of "Cut-tree" (also called Hu-Gomory tree). For example (https://stackoverflow.com/questions/25297470/igraphs-gomory-...

Given a connected graph $G=(V,E)$ with a weight function $w:E\to\mathbb{R}$ and a subset $E_0\subseteq E$ such that the subgraph $(V,E_0)$ is connected, I am looking for a sequence $E_0\subseteq E_1\...