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The s2 super-nodes are organized in s rows and s columns (as in the torus). Note that in each super-node at most one vertex can be chosen for the independent set. This chosen vertex will represent the tile chosen for this location in the torus tiling. Edges are placed between vertices in adjacent super-nodes (vertical and horizontal), to correspond to the adjacency constraints of the tiling. Speciﬁcally, let HT be the bipartite graph with |T | vertices at each side, such that there is an edge i → j iﬀ tile ti cannot be placed to the left of tile tj .
If such a subgraph does not exist, we deﬁne a(i, c) = +∞. We can then develop recursive formulas for a(i, c), starting from the leaves of T . Looking at the values of a(r, c) we can decide if such a subgraph exists in G. The complete proof of this lemma can be found in . Theorem 3. For any d ≥ 3 and any function f : N → N, MSMDd is fixedparameter tractable on Gf . Furthermore, the algorithm runs in time O((d + 2 1)f (2k) (f (2k) + 1)d n2 ). Proof. Given the input graph G = (V, E) ∈ Gf , that is, G has a bounded local tree-width and the bound is given by the function f .
840–849 (2004) 7. : Algorithmic Graph Minor Theory: Decomposition, Approximation and Coloring. In: FOCS, pp. 637–646 (2005) 8. : Traﬃc grooming in WDM networks: Past and future. IEEE Network 16(6), 46–56 (2002) 9. : Diameter and Tree-width in Minor-closed Graph Families. Algorithmica 27(3–4), 275–291 (2000) 10. : The Dense k-Subgraph Problem. Algorithmica 29(3), 410–421 (2001) 11. : On the ﬁxed-parameter intractability and tractability of multiple-interval graph properties. Manuscript (2007) 12.
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