  In the complex of curves of a closed orientable surface of genus $g,$
$\mathcal{C}(S_g),$ a preferred finite set of geodesics between any two
vertices, called \emph{efficient geodesics} introduced by Birman, Margalit, and
Menasco in \cite{birman_margalit_menasco_2016}. The main tool used to establish
the existence of efficient geodesics was a \emph{dot graph}, which recorded the
intersection pattern of a reference arc with the simple closed curves
associated with a geodesic path. The idea behind the construction was that a
geodesic that is not initially efficient contains shapes in its corresponding
dot graph.
  These shapes then correspond to surgeries that reduce the intersection with
the reference arc. In this paper, we show that the efficient geodesic algorithm
is able to be restricted to the non-separating curve complex; the proof of this
will involve analysis of the dot graph and its corresponding surgeries.
Moreover, we demonstrate that given any geodesic in the complex of curves we
may obtain an efficient geodesic whose vertices, with the possible exception of
the endpoints, are all nonseparating curves.
