Farrell Polynomials on Graphs of Bounded Tree Width

range of the weight function on F is innnite. Nevertheless, we conjecture: Conjecture 2. On graphs of tree width at most k the frame polynomials can be computed in polynomial time. It is not yet clear how the newly introduced interlace polynomials of Arratia, Bollobb as and Sorkin ABS] t into our framework. For for graphs representing link diagrams the are also computable from the Kauuman brackets, as shown in ABS]. Problem 2. Are the interlace polynomials computable in polynomial time on graphs of bounded tree width? Finally, our methods may have some bearing on the search of complete invariant for graphs of bounded tree width. In GM99], it is shown that there is a graph canonization, and hence a complete graph invariant, for graphs of bounded tree width which is computable in polynomial time. However, this canonozation is not given as a graph polynomial. Problem 3. Is there a natural family Inv(k) of polynomials which is a complete invariant for graphs of tree width at most k? The family of frame polynomials and of permanent polynomials may be good candidates. On the xed parameter complexity of graph enumeration problems deenable in monadic second order logic. 10 Proof (Outline). For the case that (G; X) is an MSOL-polynomial, we proceed as in CMR00b], but compute in the polynomial ring R F , using a unit cost computation model for operations in the ring. For the case that (G; X) is an MSOL-Farrell, we have to rework the proof of CMR00b] considerably. In this case the assumption that the covers are vertex disjoint and that the weights are given to the connected components of the cover are essential. Again we use the tree decomposition as the basis for an inductive computation, but we have to track the connected components of the cover on the way, which creates additional complications. This results from the fact, that neither the decomposition theorem a la Feferman-Vaught (as in CMR00b]), nor the automata theory approach (as in ALS91]) give us enough information on how to handle the various connected components of the F-cover. Instead, we exploit remark 1, after the deenition of k-tree decompositions. The details will be given in the full paper. 5 Limitations of the methods From theorem 1 (and the results in Mak00,Mak01]) it follows that the following polynomials can be computed in polynomial time on graphs of tree width at most k: …