An algebra lemma, Edition: version 28 Jan 2016 by Brian Osserman

By Brian Osserman

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G. [9]). This greatly simplifies the description of the algorithms, as the polynomial multiplication is actually an important computational step. Similarly, we could define the generating polynomials for independent sets, vertex covers, dominating sets, and so on. These polynomials are defined for all graphs, and some might well be worth studying for their structural properties. Definition 2. With br being the number of independent sets of size r n br xr fI (G; x) = r=0 is the independent set generating polynomial.

A main focus is on parametrized complexity of counting and evaluation problems on graphs definable in Monadic Second Order Logic [10,11,12]. For bounded tree-width these problems are solvable in polynomial time. The resulting running time is O(f (k)n4 ) with the dependence on the parameter f (k) double exponential. When our algorithm is applied to graph polynomials for graphs of bounded tree width, then f (k) is singly exponential. There is some confusion caused by a linear time algorithm for the interlace polynomial for graphs of bounded tree-width [13].

Ruling out PTAS for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM J. Comput. 36, 1025–1071 (2006) 14. : On the hardness of approximating spanners. Algorithmica 30, 432– 450 (2001) 15. : Hardness of approximation for vertexconnectivity network design problems. SIAM Journal on Computing 33, 185–199 (2004) 16. : Approximation algorithms for the Label-CoverMAX and Red-Blue Set Cover problems. J. jp Abstract. An L(2, 1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that | f (x) − f (y)| ≥ 2 if x and y are adjacent and | f (x) − f (y)| ≥ 1 if x and y are at distance 2, for all x and y in V(G).

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