## Abstract

Boolean-width is a recently introduced graph invariant. Similar to tree-width, it measures the structural complexity of graphs. Given any graph G and a decomposition of G of boolean-width k, we give algorithms solving a large class of vertex subset and vertex partitioning problems in time O*(2 ^{O(k2)}). We relate the boolean-width of a graph to its branch-width and to the boolean-width of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rank-width in [S. Oum. Rank-width is less than or equal to branch-width. Journal of Graph Theory 57(3):239-244, 2008]. For an n-vertex random graph, with a uniform edge distribution, we show that almost surely its boolean-width is Θ(log^{2} n) - setting boolean-width apart from other graph invariants - and it is easy to find a decomposition witnessing this. Combining our results gives algorithms 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*(2^{O(log4 n)}).

Original language | English |
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Title of host publication | Graph-Theoretic Concepts in Computer Science - 36th International Workshop, WG 2010, Revised Papers |

Pages | 159-170 |

Number of pages | 12 |

DOIs | |

State | Published - 2010 |

Event | 36th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2010 - Zaros, Crete, Greece Duration: 28 Jun 2010 → 30 Jun 2010 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6410 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 36th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2010 |
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Country/Territory | Greece |

City | Zaros, Crete |

Period | 28/06/10 → 30/06/10 |

### Bibliographical note

Funding Information:Supported by the Norwegian Research Council, projects PARALGO and Graph Searching.

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science