Directed graphs of entanglement two

Authors:
Erich Grädel;Łukasz Kaiser;Roman Rabinovich
Affiliations:
Mathematische Grundlagen der Informatik, RWTH Aachen University;Mathematische Grundlagen der Informatik, RWTH Aachen University;Mathematische Grundlagen der Informatik, RWTH Aachen University
Venue:
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
Year:
2009

Citing 7
Cited 2

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Entanglement and the complexity of directed graphs

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Abstract

Entanglement is a complexity measure for directed graphs that was used to show that the variable hierarchy of the propositional modal µ-calculus is strict. While graphs of entanglement zero and one are indeed very simple, some graphs of entanglement two already contain interesting nesting of cycles. This motivates our study of the class of graphs of entanglement two, as these are both simple in a sense and already complex enough for modelling certain structured systems. Undirected graphs of entanglement two were already studied by Belkhir and Santocanale and a structural decomposition for such graphs was given. We study the general case of directed graphs of entanglement two and prove that they can be decomposed as well, in a way similar to the known decompositions for tree-width, DAG-width and Kelly-width. Moreover, we show that all graphs of entanglement two have both DAG-width and Kelly-width three. Since there exist both graphs with DAGwidth three and graphs with Kelly-width three, but with arbitrary high entanglement, this confirms that graphs of entanglement two are a very basic class of graphs with cycles intertwined in an interesting way.