  We characterize the graphs formed by two-dimensional soap bubbles as being
exactly the 3-regular bridgeless planar multigraphs. Our characterization
combines a local characterization of soap bubble graphs in terms of the
curvatures of arcs meeting at common vertices, a proof that this
characterization remains invariant under Moebius transformations, an
application of Moebius invariance to prove bridgelessness, and a
Moebius-invariant power diagram of circles previously developed by the author
for its applications in graph drawing.
