Motivated by a problem about totally odd immersions of graphs, we define the odd edge-connectivity as the maximum number of edge-disjoint trails of odd length from to . It was recently discovered that can be approximated up to a constant multiplicative factor using the usual edge-connectivity between and and the minimum value of another parameter that measures "how far from a bipartite graph" the part of the graph around and 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.