  Let p be a singular point of a variety. Consider a resolution where the
preimage of p is a simple normal crossing divisor E. The combinatorial
structure of E is described by a cell complex D(E), called the dual graph or
dual complex of E. It is known that the homotopy type of D(E) depends only on
p, not on the resolution chosen.
  We prove that this homotopy type can be arbitrary. We also describe which
homotopy types can be obtained from rational singularities.
