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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New Width Parameters of Graphs
TLDR
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.
An Experimental Approach to Exact and Random Boolean-Widths and Their Comparison with Other Width Parameters
  • S. Sharmin
  • Computer Science
    The Computer Journal
  • 2021
TLDR
The results of this paper demonstrate that the boolean-width can be used not only in theory but also in practice and is competitive with other width parameters for real graphs.
On Digraph Width Measures in Parameterized Algorithmics
TLDR
New measures K-width and DAG-depth are introduced, noting that taking Kante's directed generalization of rank-width as a parameter makes many problems fixed parameter tractable and many quite different problems remain hard even on graph classes that are restricted very beyond simply having small D AG-width.
Between Treewidth and Clique-Width
TLDR
The graph parameter sm-width is considered which lies between treewidth and clique-width and it is shown that MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized by sm- width.
On the Boolean-Width of a Graph: Structure and Applications
TLDR
Algorithms are given that on input a random graph on n vertices will solve a large class of vertex subset and vertex partitioning problems in quasi-polynomial time O*(2O(log4 n)).
Are there any good digraph width measures?
Digraph width measures in parameterized algorithmics
Mim-Width I. Induced path problems
Fast algorithms for vertex subset and vertex partitioning problems on graphs of low boolean-width ⋆
TLDR
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.
Boolean-Width of Graphs
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…
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