Structural complexity 1
Tree automata, Mu-Calculus and determinacy
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Graph searching and a min-max theorem for tree-width
Journal of Combinatorial Theory Series B
Infinite games on finitely coloured graphs with applications to automata on infinite trees
Theoretical Computer Science
A partial k-arboretum of graphs with bounded treewidth
Theoretical Computer Science
Journal of the ACM (JACM)
Journal of Combinatorial Theory Series B
Small Progress Measures for Solving Parity Games
STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
On the Variable Hierarchy of the Modal µ-Calculus
CSL '02 Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
DAG-width: connectivity measure for directed graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Digraph measures: Kelly decompositions, games, and orderings
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
The Variable Hierarchy of the μ-Calculus Is Strict
Theory of Computing Systems
D-width, metric embedding, and their connections
D-width, metric embedding, and their connections
Digraph measures: Kelly decompositions, games, and orderings
Theoretical Computer Science
The Descriptive Complexity of Parity Games
CSL '08 Proceedings of the 22nd international workshop on Computer Science Logic
Digraph Decompositions and Monotonicity in Digraph Searching
Graph-Theoretic Concepts in Computer Science
Space-bounded reducibility among combinatorial problems
Journal of Computer and System Sciences
Undirected graphs of entanglement 2
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
Directed graphs of entanglement two
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
The dag-width of directed graphs
Journal of Combinatorial Theory Series B
Entropy and heterogeneity measures for directed graphs
SIMBAD'13 Proceedings of the Second international conference on Similarity-Based Pattern Recognition
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Entanglement is a parameter for the complexity of finite directed graphs that measures to what extent the cycles of the graph are intertwined. It is defined by way of a game similar in spirit to the cops and robber games used to describe treewidth, directed treewidth, and hypertree width. Nevertheless, on many classes of graphs, there are significant differences between entanglement and the various incarnations of treewidth. Entanglement is intimately related with the computational and descriptive complexity of the modal @m-calculus. The number of fixed-point variables needed to describe a finite graph up to bisimulation is captured by its entanglement. This plays a crucial role in the proof that the variable hierarchy of the @m-calculus is strict. We study complexity issues for entanglement and compare it to other structural parameters of directed graphs. One of our main results is that parity games of bounded entanglement can be solved in polynomial time. Specifically, we establish that the complexity of solving a parity game can be parametrised in terms of the minimal entanglement of subgames induced by a winning strategy. Furthermore, we discuss the case of graphs of entanglement two. While graphs of entanglement zero and one are very simple, graphs of entanglement two allow arbitrary nesting of cycles, and they form a sufficiently rich class for modelling relevant classes of structured systems. We provide characterisations of this class, and propose decomposition notions similar to the ones for treewidth, DAG-width, and Kelly-width.