  We study graphs which admit an acyclic orientation that contains an
out-branching and in-branching which are arc-disjoint (such an orientation is
called {\bf good}). A {\bf 2T-graph} is a graph whose edge set can be
decomposed into two edge-disjoint spanning trees. Clearly a graph has a good
orientation if and only if it contains a spanning 2T-graph with a good
orientation, implying that 2T-graphs play a central role. Vertex-minimal
2T-graphs with at least two vertices, also known as {\bf generic circuits},
play an important role in rigidity theory for graphs. It was shown in
\cite{bangGOpaper} that every generic circuit has a good orientation. Using
this, several results on good orientations of 2T-graphs were obtained in
\cite{bangGOpaper}. It is an open problem whether there exist a polynomial
algorithm for deciding whether a given 2T-graph has a good orientation. In
\cite{bangGOpaper} complex constructions of 2T-graphs with no good orientation
were given, indicating that the problem might be very difficult. In this paper
we focus on so-called {\bf quartics} which are 2T-graphs where every vertex has
degree 3 or 4. We identify a sufficient condition for a quartic to have a good
orientation, give a polynomial algorithm to recognize quartics satisfying the
condition and a polynomial algorithm to produce such an orientation when this
condition is met. As a consequence of these results we prove that every
4-regular and 4-connected graph has a good orientation. \iffalse We also
provide evidence that even for quartics it may be difficult to find a
characterization of those instances which have a good orientation.\fi We also
show that every graph on $n\geq 8$ vertices and of minimum degree at least
$\lfloor{}n/2\rfloor$ has a good orientation. Finally we pose a number of open
problems.
