# Width Parameters Beyond Tree-width and their Applications

@article{Hlinn2008WidthPB, title={Width Parameters Beyond Tree-width and their Applications}, author={Petr Hliněn{\'y} and Sang-il Oum and Detlef Seese and Georg Gottlob}, journal={Comput. J.}, year={2008}, volume={51}, pages={326-362} }

Besides the very successful concept of tree-width (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our…

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## 181 Citations

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The notion of branch decompositions of a set function introduced by Robert-son and Seymour is used to introduce three new graph parameters, boolean-width, max-imum matching-width (MM-width) and maximum induced matching- width (MIM-width), which are compared to existing graph parameters by comparing partial orders of width parameters.

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Which are the graph problems having the same behaviour of boolean-width and rankwidth, and are extended to a large class of NP-hard vertex subset and vertex partitioning problems by giving algorithms that are FPT when parameterized by either Boolean-width, rank-width or clique-width.

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We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when…

## References

SHOWING 1-10 OF 324 REFERENCES

A Parametrized Algorithm for Matroid Branch-Width

- Computer ScienceSIAM J. Comput.
- 2005

An algorithm is presented that, for a given matroid M of bounded branch-width t, finds a branch decomposition of M of width at most 3t in cubic time, and it is shown that the branch- width of M is a uniformly fixed-parameter tractable problem.

On the Relationship between Clique-Width and Treewidth

- MathematicsWG
- 2001

There is an exponential lower bound on the relationship between treewidth and clique-width and it is shown that there is a polynomial time algorithm for any graph problem expressible in Monadic Second Order Logic.

Hypergraphs in Model Checking: Acyclicity and Hypertree-Width versus Clique-Width

- Computer ScienceICALP
- 2001

This paper provides a detailed comparison of the expressive power of several restrictions on the logical formulae and on the structures under consideration, in order to guarantee the tractability of this decision problem.

Approximating clique-width and branch-width

- Computer Science, MathematicsJ. Comb. Theory, Ser. B
- 2006

New Graph Classes of Bounded Clique-Width

- MathematicsTheory of Computing Systems
- 2004

This work gives a complete classification of all graph classes defined by forbidden one-vertex extensions of the P4 with respect to bounded clique-width and extends and improves recent structural and complexity results in a systematic way.

Approximating Rank-Width and Clique-Width Quickly

- Computer ScienceWG
- 2005

Two separate algorithms of this kind with faster running time are developed, one of which is a O(|V|4)-time algorithm with f(k) = 3k + 1 by constructing a subroutine for the previous algorithm; this may now avoid using general submodular function minimization algorithms used by Seymour and the author.

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width

- Mathematics, Computer ScienceTheory of Computing Systems
- 2000

It is proved that this is also the case for graphs of clique-width at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications.

On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

- Computer Science, MathematicsDiscret. Appl. Math.
- 2001

Branch-width, parse trees, and monadic second-order logic for matroids

- MathematicsJ. Comb. Theory, Ser. B
- 2006

On the Clique-Width of Some Perfect Graph Classes

- MathematicsInt. J. Found. Comput. Sci.
- 2000

The clique–width of perfect graph classes is studied to see the border within the hierarchy of perfect graphs between classes whose clique-width is bounded and classes whoseClique– width is unbounded.