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Inspired by the study of community structure in connection networks, weintroduce the graph polynomial $Q(G;x,y)$, the bivariate generating functionwhich counts the number of connected components in induced subgraphs.  We give a recursive definition of $Q(G;x,y)$ using vertex deletion, vertexcontraction and deletion of a vertex together with its neighborhood and prove auniversality property. We relate $Q(G;x,y)$ to other known graph invariants andgraph polynomials, among them partition functions, the Tutte polynomial, theindependence and matching polynomials, and the universal edge eliminationpolynomial 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 andUlam, discuss its use in computing residual connectedness reliability. Finallywe show that the computation of $Q(G;x,y)$ is $\sharp \mathbf{P}$-hard, butFixed Parameter Tractable for graphs of bounded tree-width and clique-width.

The Enumeration of Vertex Induced Subgraphs with respect to the Number of Components