Recurrence relations and splitting formulas for the domination   polynomial
  The domination polynomial D(G,x) of a graph G is the generating function of its dominating sets. We prove that D(G,x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs and for various special cases. We give splitting formulas for D(G,x) based on articulation vertices, and more generally, on splitting sets of vertices. 
Tomer Kotek, James Preen, Frank Simon, Peter Tittmann and Martin   Trinks
math.CO