  The $c_2$ invariant is an arithmetic graph invariant defined by Schnetz. It
is useful for understanding Feynman periods. Brown and Schnetz conjectured that
the $c_2$ invariant has a particular symmetry known as completion invariance.
This paper will prove completion invariance of the $c_2$ invariant in the case
that $p=2$ and the completed graph has an odd number of vertices.
  The methods involve enumerating certain edge bipartitions of graphs; two
different constructions are needed.
