  It is known that graphs cellularly embedded into surfaces are equivalent to
ribbon graphs. In this work, we generalize this statement to broader classes of
graphs and surfaces. Half-edge graphs extend abstract graphs and are useful in
quantum field theory in physics. On the other hand, ribbon graphs with
half-edges generalize ribbon graphs and appear in a different type of field
theory emanating from matrix models. We then give a sense of embeddings of
half-edge graphs in punctured surfaces and determine (minimal/maximal)
conditions for an equivalence between these embeddings and half-edge ribbon
graphs. Given some assumptions on the embedding, the geometric dual of a
cellularly embedded half-edge graph is also identified.
