# Decomposition trees: Structured graph representation and efficient algorithms

- 18 Citations
- 158 Downloads

## Abstract

A data structure, *decomposition trees*, is introduced, which enables graphs to be represented in a certain structured way, and which leads to simple, recursive algorithms for many difficult graph problems.

For a number of *NP*-complete problems these algorithms are shown to run in linear time on decomposition trees with bounded *label size*. Furthermore it is shown that for those graphs which have decomposition tree representations with bounded label size such a representation can be constructed in polynomial time.

Put together, these algorithms solve a number of *NP*-complete problems in polynomial time on many graph classes, including all those graph languages that can be generated by any sort of context-free graph grammars, e.g., (hyper-)edge replacement grammars.

## Preview

Unable to display preview. Download preview PDF.

## References

- [Ar85]S. Arnborg,
*Efficient algorithms for combinatorial problems on graphs with bounded decomposability — a survey*. BIT 25 (1985), pp.2–23.Google Scholar - [ACP87]S. Arnborg, D. Corneil, A. Proskurowski,
*Complexity of finding embeddings in a k-tree*. SIAM J.Alg.Disc.Meth. 8 (1987), pp.277–284.Google Scholar - [ALS87]S.Arnborg, J.Lagergren, D.Seese,
*Which problems are easy for tree-decomposable graphs?*Draft, November 1987.Google Scholar - [GJ79]M. Garey, D. Johnson,
*Computers and intractability*. Freeman, N.Y., 1979.Google Scholar - [HK87a]A. Habel, H.-J. Kreowski,
*Some structural aspects of hypergraph languages generated by hyperedge replacement*. LNCS 247 (1987), pp.207–219.Google Scholar - [HK87b]A.Habel, H.-J.Kreowski,
*May we introduce to you: Hyperedge replacement*. Proc. 3^{rd}international workshop on graph grammars and their applications to Computer Science, to appear.Google Scholar - [Kr86]H.-J.Kreowski,
*Rule trees can help to escape hard graph problems*. Preprint, Universität Bremen, 1986.Google Scholar - [LW87]K.-J. Lange, E. Welzl,
*String grammars with disconnection*. Discr.Appl.Math. 16 (1987), pp.17–30.Google Scholar - [Le86]T. Lengauer,
*Efficient algorithms for finding minimum spanning forests of hierarchically defined graphs*. LNCS 216 (1986), pp.153–170.Google Scholar - [RS86]N. Robertson, P. Seymour,
*Graph minors. II. Algorithmic aspects of tree-width*. J.Algorithms 7 (1986), pp.309–322.Google Scholar - [RW86]G. Rozenberg, E. Welzl,
*Boundary NLC grammars — basic definitions, normal forms and complexity*. Information and Control 69 (1986), pp.136–167.Google Scholar - [Sl82]A.O. Slisenko,
*Context-free graph grammars as a tool for describing polynomial-time subclasses of hard problems*. Inf.Proc.Let. 14 (1982), pp.52–56.Google Scholar