We find that hyperelliptic graphs in the sense of Baker and Norine are planar and examine connections between the gonality and orientable genus of a graph. 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 ≥ 0, trigonal graphs of orientable genus g, and give analogues for graphs of higher gonality.

Much of this paper was developed in conversation with Spencer Backman. We thank him for numerous ideas. We also thank the anonymous referee for helpful comments.