When Do Gomory-Hu Subtrees Exist?

Abstract

Gomory-Hu (GH) Trees are a classical sparsification technique for graph connectivity. It is one of the fundamental models in combinatorial optimization which also continually finds new applications, most recently in social network analysis. For any edge-capacitated undirected graph $G=(V,E)$ and any subset of {\em terminals} $Z \subseteq V$, a Gomory-Hu Tree is an edge-capacitated tree $T=(Z,E(T))$ such that for every $u,v \in Z$, the value of the minimum capacity $uv$ cut in $G$ is the same as in $T$. Moreover, the minimum cuts in $T$ directly identify (in a certain way) those in $G$. It is well-known that we may not always find a GH tree which is a subgraph of $G$. For instance, every GH tree for the vertices 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 \emph{terminal-$K_{2,3}$ minor}. That is, no $K_{2,3}$ minor whose vertices correspond to terminals in $Z$. We also show that the family of pairs $(G,Z)$ which forbid such $K_{2,3}$ "$Z$-minors" arises, roughly speaking, from so-called Okamura-Seymour instances. More precisely, they are subgraphs of {\em $Z$-webs}. A $Z$-web is built from planar graphs with one outside face which contains all the terminals and each inner face is a triangle which may contain an arbitrary graph. This characterization yields an additional consequence for multiflow problems. Fix a graph $G$ and a subset $Z \subseteq V(G)$ of terminals. Call $(G,Z)$ {\em cut-sufficient} if the cut condition is sufficient to characterize the existence of a multiflow for any demands between vertices 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.