  We consider the thickness $\theta (G))$ and outerthickness $\theta _o(G)$ of
a graph G in terms of its orientable and nonorientable genus. Dean and
Hutchinson provided upper bounds for thickness of graphs in terms of their
orientable genus. More recently, Concalves proved that the outerthickness of
any planar graph is at most 2. In this paper, we apply the method of deleting
spanning disks of embeddings to approximate the thickness and outerthickness of
graphs. We first obtain better upper bounds for thickness. We then use a
similar approach to provide upper bounds for outerthickness of graphs in terms
of their orientable and nonorientable genera. Finally we show that the
outerthickness of the torus (the maximum outerthickness of all toroidal graphs)
is 3. We also show that all graphs embeddable in the double torus have
thickness at most 3 and outerthickness at most 5.
