F. Gurski and E. Wanke. The tree-width of clique-width bounded graphs without K n,n .InProceedings of Graph-Theoretical Concepts in Computer Science, volume 1938 of LNCS, pages 196--205. Springer-Verlag, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....width of a graph G is the minimal number of colors that is needed in such an expression de ning G. The following propositions relate the notions of clique width and tree width. Proposition 4. CO00] Let G be a graph. If G has bounded tree width, then G has bounded clique width. Proposition 5. [GW00] Let G be a graph. If G is of bounded clique width and there is an n 2 N such that K n;n is not a subgraph of G, then G is of bounded tree width. We will generate graphs using ground tree rewriting systems. These are introduced in the next section. Here we de ne nite ranked trees, the objects ....

Frank Gurski and Egon Wanke. The tree-width of clique-width bounded graphs without Kn;n . In Proceedings of WG 2000, volume 1928 of LNCS, pages 196-205. Springer-Verlag, 2000.

....from the results in Chapter 4 that the NLCT width of a graph G is bounded by log n times the NLC width. Here n is the number of vertices in G. Bounded NLC width clique width does not in general imply bounded treewidth. However, for graphs without large complete bipartite subgraphs, it does [16]. Results have also been obtained for certain families of graphs with restrictions on the number of induced P 4 subgraphs [21] Finally, distance hereditary graphs have clique width at most 3, whereas unit interval graphs and permutation graphs have unbounded clique width [15] Chapter 3 NLC 2 ....

Frank Gurski and Egon Wanke. The tree-width of clique-width bounded graphs without K n,n .InProc. 26th Int. Workshop on Graph-Theoretic Concepts in Computer Science, volume

....proves proposition 13. A more informative proof of proposition 13 is as follows: Proposition 33 Let K 2 be the closure under induced subgraphs of the class f(Grid n ) 2 n : n 1g. K 2 has unbounded clique width and is MSOL polynomial. 12 PROOF. K 2 contains only planar graphs, hence, by [GW00], if the clique width of K 2 were bounded by k, so its tree width would be bounded by 6k Gamma 1. But the grids Grid n are topological minors of graphs in K 2 and have tree width O(n) Hence the clique width of K 2 is unbounded. To show that K 2 is MSOL polynomial we proceed as in the proof of ....

....unbounded. To show that K 2 is MSOL polynomial we proceed as in the proof of proposition 32. Q.E.D. This proves propositions 12 and 13. 6 Cliquewidth vs treewidth Here we discuss under what closure condtions a class of graphs of cliquewidth at most k has bounded treewidth. F. Gurski and E. Wanke [GW00] proved among other properties of graphs of cliquewidth k the following: Proposition 34 Let G be a graph of cliquewidth at most k. i) If G is planar, the G has treewidth at most 6k Gamma 1. ii) If G has degree at most d, then G has treewidth at most 3kd Gamma 1. We now want to prove ....

F. Gurski and E. Wanke. The tree-width of clique-width bounded graphs without k n;n . Lecture Notes in Computer Science,

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F. Gurski and E. Wanke. The tree-width of clique-width bounded graphs without K n,n .InProceedings of Graph-Theoretical Concepts in Computer Science, volume 1938 of LNCS, pages 196--205. Springer-Verlag, 2000.

....at most n r, if 2 r n r, and NLC width at most d n 2 e, see [Joh98] Every graph of tree width at most k has clique width at most 3 2 k 1 , see [CR01] and NLC width at most 2 k 1 1, see [Wan94] The graphs of clique width at most 2 or NLC width 1 do not have bounded tree width. In [GW00], it is shown that every graph of clique width or NLC width at most k which does not contain the complete bipartite graph K n;n for some n 1 as a subgraph has tree width at most 3k(n 1) 1. The recognition problem for graphs of clique width or NLCwidth at most k is still open for k 4 and k 3, ....

F. Gurski and E. Wanke. The tree-width of clique-width bounded graphs without Kn;n . In Proceedings of Graph-Theoretical Concepts in Computer Science, volume

....at most n r, if 2 r n r, and NLC width at most d n 2 e, see [Joh98] Every graph of tree width at most k has clique width at most 3 2 k 1 , see [CR01] and NLC width at most 2 k 1 1, see [Wan94] The graphs of clique width at most 2 or NLC width 1 do not have bounded tree width. In [GW00], it is shown that every graph of clique width or NLC width at most k which does not contain the complete bipartite graph K n;n for some n 1 as a subgraph has tree width at most 3k(n 1) 1. The recognition problem for graphs of clique width or NLCwidth at most k is still open for k 4 and k 3, ....

F. Gurski and E. Wanke. The tree-width of clique-width bounded graphs without Kn;n . In Proceedings of Graph-Theoretical Concepts in Computer Science, volume

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