### Abstract

One of the best-known methods for decomposing graphs is the method of tree-decompositions introduced by Robertson and Seymour. Many NP-hard problems become polynomially soblvable if restricted to instances whose underlying graph structure has bounded treewidth. The notion of treewidth can be straightforwardly extended to hypergraphs by simply considering the treewidth of their primal graphs or, alteratively, of their incidence graphs. However, doing so comes along with a loss of information on the structure of a hypergraph with the effect that many polynomially solvable problems cannot be recognized as such because the treewidth of the underlying hypergraphs is unbounded. In particular, the treewidth of the class of acyclic hypergraphs is unbounded. In this talk, I will describe more appropriate measures for hypergraph acyclicity, and, in particular, the method of hypertree decompositions and the associated concept of hypertree width. After giving general results on hypertree decompositions, I will report on game-theoretic characterizations of hypergraph decomposition methods, and give a survey on recent results.

### Related Links

- http://www.dbai.tuwien.ac.at/proj/hypertree/ - Vienna Hypertree Project Page
- http://wwwinfo.deis.unical.it/~frank/Hypertrees/ - Hypertree Decomposition Homepage