Binary matroids and degree-boundedness for pivot-minors

We prove that for every bipartite graph H and positive integer s, the class of Ks,s-subgraph-free graphs excluding H as a pivot-minor has bounded average degree. Our proof relies on the announced binary matroid structure theorem of Geelen, Gerards, and Whittle. Along the way, we also prove that every Ks,t-free bipartite circle graph with s≤ t has a vertex of degree at most max\2s-2, t-1\ and provide examples showing that this is tight.