Number of holes in a polygon
Python, 71

What follows is a program (not a function) that calculates the desired number. 

len(set().union(*(map(frozenset,zip(t,t[1:]+t))for t in T)))-len(L+T)+1


Example usage:

&gt;&gt;&gt; L = ((0,0),(1,0),(2,0),(2,1),(2,2),(1,2),(0,2),(0,1),(.5,.5),(1.5,.5),(1.5,1.5),(.5,1.5))
&gt;&gt;&gt; T = ((5,6,11),(5,10,11),(4,5,10),(3,8,10),(2,3,9),(2,8,9),(1,2,8),(0,1,8),(0,8,11),(0,7,11),(6,7,11),(3,4,10))
&gt;&gt;&gt; len(set().union(*(map(frozenset,zip(t,t[1:]+t))for t in T)))-len(L+T)+1
2