On graph invariants given by linear recurrence relations
Journal of Combinatorial Theory Series B
Chromatic polynomials of homeomorphism classes of graphs
Discrete Mathematics - Special issue on selected papers in honor of Henry W. Gould
Upper bounds to the clique width of graphs
Discrete Applied Mathematics
Chain polynomials and Tutte polynomials
Discrete Mathematics
The vertex-cover polynomial of a graph
Discrete Mathematics
A Tutte Polynomial for Coloured Graphs
Combinatorics, Probability and Computing
Coloured Tutte polynomials and Kauffman brackets for graphs of bounded tree width
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
On the colored Tutte polynomial of a graph of bounded treewidth
Discrete Applied Mathematics
Counting truth assignments of formulas of bounded tree-width or clique-width
Discrete Applied Mathematics
From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials
Theory of Computing Systems
Complexity of the Bollobás-Riordan polynomial: exceptional points and uniform reductions
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
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
Complexity of the cover polynomial
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
The enumeration of vertex induced subgraphs with respect to the number of components
European Journal of Combinatorics
Distinguishing graphs by their left and right homomorphism profiles
European Journal of Combinatorics
On the roots of edge cover polynomials of graphs
European Journal of Combinatorics
On the edge cover polynomial of a graph
European Journal of Combinatorics
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K. Dohmen, A. Ponitz and P. Tittmann [K. Dohmen, A. Ponitz, P. Tittmann, A new two-variable generalization of the chromatic polynomial, Discrete Mathematics and Theoretical Computer Science 6 (2003), 69-90], introduced a bivariate generalization of the chromatic polynomial P(G,x,y) which subsumes also the independent set polynomial of I. Gutman and F. Harary [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Mathematicae 24 (1983), 97-106] and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little [F.M. Dong, M.D. Hendy, K.L. Teo, and C.H.C. Little, The vertex-cover polynomial of a graph, Discrete Mathematics 250 (2002), 71-78]. We first show that P(G,x,y) has a recursive definition with respect to three kinds of edge eliminations: edge deletion, edge contraction, and edge extraction, i.e. deletion of an edge together with its endpoints. Like in the case of deletion and contraction only [J.G. Oxley and D.J.A. Welsh, The Tutte polynomial and percolation, in: J.A. Bundy, U.S.R. Murty (Eds.), Graph Theory and Related Topics, Academic Press, London, 1979, pp. 329-339] it turns out that there is a most general, or as they call it, a universal polynomial satisfying such recurrence relations with respect to the three kinds of edge eliminations, which we call @x(G,x,y,z). We show that the new polynomial simultaneously generalizes, P(G,x,y), as well as the Tutte polynomial and the matching polynomial, We also give an explicit definition of @x(G,x,y,z) using a subset expansion formula. We also show that @x(G,x,y,z) can be viewed as a partition function, using counting of weighted graph homomorphisms. Furthermore, we expand this result to edge-labeled graphs as was done for the Tutte polynomial by T. Zaslavsky [T. Zaslavsky, Strong Tutte functions of matroids and graphs, Trans. Amer. Math. Soc. 334 (1992), 317-347] and by B. Bollobas and O. Riordan [B. Bollobas, O. Riordan, A Tutte polynomial for coloured graphs, Combinatorics, Probability and Computing 8 (1999), 45-94]. The edge-labeled polynomial @x"l"a"b(G,x,y,z,t@?) also generalizes the chain polynomial of R.C. Read and E.G. Whitehead Jr. [R.C. Read, E.G. Whitehead Jr., Chromatic polynomials of homeomorphism classes of graphs, Discrete Mathematics 204 (1999), 337-356]. Finally, we discuss the complexity of computing @x(G,x,y,z).