Abstract
Inspired by the study of community structure in connection networks, we introduce the graph polynomial \(Q\left( G;x,y\right)\), as a bivariate generating function which counts the number of connected components in induced subgraphs. We analyze the features of the new polynomial. First, we re-define it as a subset expansion formula. Second, we give a recursive definition of \(Q\left( G;x,y\right)\) using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood, and prove a universality property. We relate \(Q\left( G;x,y\right)\) to the universal edge elimination polynomial introduced by I. Averbouch, B. Godlin and J.A. Makowsky (2008), which subsumes other known graph invariants and graph polynomials, among them the Tutte polynomial, the independence and matching polynomials, and the bivariate extension of the chromatic polynomial introduced by K. Dohmen, A. Pönitz, and P. Tittmann (2003). Finally we show that the computation of \(Q\left( G;x,y\right)\) is \(\sharp \mathbf{P}\)-hard, but Fixed Parameter Tractable for graphs of bounded tree-width and clique-width.
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Averbouch, I., Makowsky, J.A., Tittmann, P. (2010). A Graph Polynomial Arising from Community Structure (Extended Abstract). In: Paul, C., Habib, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2009. Lecture Notes in Computer Science, vol 5911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11409-0_3
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