When Do Gomory-Hu Subtrees Exist?

Gomory-Hu (GH) Trees are a classical sparsification technique for graph connectivity. It is also a fundamental model in combinatorial optimization which finds new applications, for instance , finding highly-connected communities within (social) networks. For any edge-capacitated undirected graph G = (V, E) and any subset of terminals Z ⊆ V , a Gomory-Hu Tree is an edge-capacitated tree T = (Z, E(T)) such that for every u, v ∈ Z, the value of the minimum capacity uv cut in G is the same as in T. It is well-known that we may not always find a GH tree which is a subgraph (or minor if Z = V) of G. For instance, every GH tree for the nodes of K 3,3 is a 5-star. We characterize those graph and terminal pairs (G, Z) which always admit such a tree. We show that these are the graphs which have no terminal-K 2,3 minor, that is, a K 2,3 minor each of whose nodes corresponds to a terminal of Z. We then show that the pairs (G, Z) which forbid such K 2,3 "Z-minors" arises, roughly speaking, from so-called Okamura-Seymour instances. These are planar graphs where the outside face contains all the terminals. This characterization yields an unexpected consequence for multiflow problems which extends an earlier line of inquiry due to Lomonosov and Seymour. Fix a graph G and subset Z ⊆ V (G) of terminals. Call (G, Z) cut-sufficient if the cut condition is sufficient to characterize the existence of a multiflow for any demands between nodes in Z, and any edge capacities on G. Then (G, Z) is cut-sufficient if and only if it is terminal-K 2,3 free.