Motivated by a problem about totally odd immersions of graphs, we define the *odd edge-connectivity* *λ _{o}(u, v)* as the maximum number of edge-disjoint trails of odd length from

*u*to

*υ*. It was recently discovered that

*λ*can be approximated up to a constant multiplicative factor using the usual edge-connectivity between u and v and the minimum value of another parameter that measures “how far from a bipartite graph” the part of the graph around

_{o}(u, v)*u*and

*v*is.

In this paper, we formalize this second ingredient and call it the *perimeter*. We prove that perimeter is a submodular function on the vertex-sets of a graph. Using this fact, we obtain a version of the Gomory–Hu Theorem in which minimum edge-cuts are replaced by sets of minimum perimeter. We construct (in polynomial time) a rooted forest structure, analogous to the Gomory-Hu tree of a graph, which encodes a collection of minimum-perimeter vertex-sets. Although the classical Gomory-Hu Theorem extends to arbitrary *symmetric* submodular functions, our result is novel and indicates a possibility for further generalizations.

These results have significant implications for the study of path and trail systems with parity constraints. We present two such applications: an efficient data structure for storing approximate odd edge-connectivities for all pairs of vertices in a graph, and a rough structure theorem for graphs with no “totally odd” immersion of a large complete graph.

A submodular measure and approximate Gomory-Hu theorem for packing odd trails. Ross Churchley and Bojan Mohar. *Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018),* 210–218. January 2018.