# No nice generalization of Gomory-Hu tree

Gomory-Hu tree of a graph G is a weighted graph H on the same set of vertices. It has the two nice properties.

1. It is a tree.
2. For every s,t, there exist a min-st-cut in H that is also a min-st-cut in G.

We could ask for a generalization. Given a graph G, can we find some nice graph H such that

For every S,T such that 1\leq |S|,|T|\leq k and S\cap T=\emptyset, there is a min-ST-cut in H that is a min-ST-cut in G.

We recover Gomory-Hu tree when k=1. There were some consideration of a problem similar to this [1].

There are various useful definition of nice , for example sparse or bounded treewidth. Likely none of them would apply, because we can show that for k=2, there are graphs where H is almost the complete graph.

Let G be the complete graph on n vertices, and each edge has capacity 1. Let f(S) be the sum of the capacity of edges going across S. There is a unique ST-min cut for every |T|=1 and |S|=2. Hence this shows that f(\{v\}) = n-1 for every v\in V. If |S|=|T|=2, there are exactly two min ST-cuts, either S or T in G. Assume that T is not a min ST-cut in H, then S has to be the min ST-cut in H. Therefore for all \{s_1,s_2\} such that s_1,s_2\not \in T, we have f(\{s_1,s_2\})=2n-4. Because f(\{s_1\})=f(\{s_2\})=n-1, the edge between s_1 and s_2 has capacity 1. The complete graph on V\setminus T has to be a subgraph of H.

# Reference

[1] R. Chitnis, L. Kamma, R. Krauthgamer, Tight Bounds for Gomory-Hu-like Cut Counting, ArXiv E-Prints. (2015).

Posted by Chao Xu on 2016-02-01.
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