Treewidth and minimum fill-in on permutation graphs in linear time

• Published in: Theoretical computer science Vol. 411; no. 40; pp. 3685 - 3700
• Main Author: Meister, Daniel
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier B.V 06.09.2010; Elsevier
• Subjects: Algorithms; Applied sciences; C (programming language); Complement; Computation; Computer science; control theory; systems; Exact sciences and technology; Graph algorithms; Graphs; Information retrieval. Graph; Interval graphs; Linear time; Minimal triangulation; Miscellaneous; Permutation graphs; Permutations; Representations; Theoretical computing; Treewidth; Triangulation; Interval graphs; Linear time; Graph algorithms; Permutation graphs; Treewidth; Minimal triangulation; Interval graph; Triangulation; Computer theory; Permutation graph; Computing; Minimal graph; Subgraph; Linear graph; Graph algorithm
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2010.06.017

Permutation graphs form a well-studied subclass of cocomparability graphs. Permutation graphs are the cocomparability graphs whose complements are also cocomparability graphs. A triangulation of a graph  G is a graph  H that is obtained by adding edges to G to make it chordal. If no triangulation of G is a proper subgraph of H then H is called a minimal triangulation. The main theoretical result of the paper is a characterisation of the minimal triangulations of a permutation graph, that also leads to a succinct and linear-time computable representation of the set of minimal triangulations. We apply this representation to devise linear-time algorithms for various minimal triangulation problems on permutation graphs, in particular, we give linear-time algorithms for computing treewidth and minimum fill-in on permutation graphs.