Abstract
We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and edge extraction, i.e., deletion of edges together with their end points. Like in the case of deletion and contraction only (J.G. Oxley and D.J.A. Welsh 1979), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call ξ(G,x,y,z). We show that the new polynomial simultaneously generalizes the Tutte polynomial, the matching polynomial, and the recent generalization of the chromatic polynomial proposed by K.Dohmen, A.Pönitz and P.Tittman (2003), including also the independent set polynomial of I. Gutman and F. Harary, (1983) and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little (2002). We give three definitions of the new polynomial: first, the most general recursive definition, second, an explicit one, using a set expansion formula, and finally, a partition function, using counting of weighted graph homomorphisms. We prove the equivalence of the three definitions. Finally, we discuss the complexity of computing ξ(G,x,y,z).
Partially supported by the Israel Science Foundation for the project “Model Theoretic Interpretations of Counting Functions” (2007–2010) and by a grant of the Fund for Promotion of Research of the Technion–Israel Institute of Technology.
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Averbouch, I., Godlin, B., Makowsky, J.A. (2008). A Most General Edge Elimination Polynomial. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2008. Lecture Notes in Computer Science, vol 5344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92248-3_4
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