# Decomposition trees: Structured graph representation and efficient algorithms

## 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.

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