On graph invariants given by linear recurrence relations
Journal of Combinatorial Theory Series B
Upper bounds to the clique width of graphs
Discrete Applied Mathematics
The vertex-cover polynomial of a graph
Discrete Mathematics
A Tutte Polynomial for Coloured Graphs
Combinatorics, Probability and Computing
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Counting truth assignments of formulas of bounded tree-width or clique-width
Discrete Applied Mathematics
Coloured Tutte polynomials and Kauffman brackets for graphs of bounded tree width
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Note: On the colored Tutte polynomial of a graph of bounded treewidth
Discrete Applied Mathematics
Computing graph polynomials on graphs of bounded clique-width
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Computing the tutte polynomial on graphs of bounded clique-width
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Parameterized Complexity
The enumeration of vertex induced subgraphs with respect to the number of components
European Journal of Combinatorics
The complexity of the cover polynomials for planar graphs of bounded degree
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
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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 *** (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 *** (G ,x ,y ,z ).