  The fundamental role of on-shell diagrams in quantum field theory has been
recently recognized. On-shell diagrams, or equivalently bipartite graphs,
provide a natural bridge connecting gauge theory to powerful mathematical
structures such as the Grassmannian. We perform a detailed investigation of the
combinatorial and geometric objects associated to these graphs. We mainly focus
on their relation to polytopes and toric geometry, the Grassmannian and its
stratification. Our work extends the current understanding of these connections
along several important fronts, most notably eliminating restrictions imposed
by planarity, positivity, reducibility and edge removability. We illustrate our
ideas with several explicit examples and introduce concrete methods that
considerably simplify computations. We consider it highly likely that the
structures unveiled in this article will arise in the on-shell study of
scattering amplitudes beyond the planar limit. Our results can be conversely
regarded as an expansion in the understanding of the Grassmannian in terms of
bipartite graphs.
