  We examine connections between the gonality, treewidth, and orientable genus
of a graph. Especially, we find that hyperelliptic graphs in the sense of Baker
and Norine are planar. We give a notion of a bielliptic graph and show that
each of these must embed into a closed orientable surface of genus one. We also
find, for all $g\ge 0$, trigonal graphs of treewidth 3 and orientable genus
$g$, and give analogues for graphs of higher gonality.
