  Inspired by the study of community structure in connection networks, we
introduce the graph polynomial $Q(G;x,y)$, the bivariate generating function
which counts the number of connected components in induced subgraphs.
  We give a recursive definition of $Q(G;x,y)$ using vertex deletion, vertex
contraction and deletion of a vertex together with its neighborhood and prove a
universality property. We relate $Q(G;x,y)$ to other known graph invariants and
graph polynomials, among them partition functions, the Tutte polynomial, the
independence and matching polynomials, and the universal edge elimination
polynomial introduced by I. Averbouch, B. Godlin and J.A. Makowsky (2008).
  We show that $Q(G;x,y)$ is vertex reconstructible in the sense of Kelly and
Ulam, discuss its use in computing residual connectedness reliability. Finally
we show that the computation of $Q(G;x,y)$ is $\sharp \mathbf{P}$-hard, but
Fixed Parameter Tractable for graphs of bounded tree-width and clique-width.
