  Let $G$ be a graph such that, whenever two vertices $x$ and $y$ of $G$ are
joined by three internally disjoint paths, $x$ and $y$ are adjacent. Jamison
and Mulder determined that the set of such graphs coincides with the set of
graphs that can be built from cycles and complete graphs via 1-sums and
parallel connections. This paper proves an analogous result for binary
matroids.
