  The obstruction for application of effective techniques like denominator
reduction for the computation of the $c_2$ invariant of Feynman graphs in
general is the absence of a 3-valent vertex for the initial steps. In this
paper such a formula for a 4-valent vertex is derived. The formula allows to
compute the $c_2$ invariant of new graphs, for instance, 4-regular graphs with
small loop number.
