The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
Easy problems for tree-decomposable graphs
Journal of Algorithms
Graph rewriting: an algebraic and logic approach
Handbook of theoretical computer science (vol. B)
Monadic second-order evaluations on tree-decomposable graphs
Theoretical Computer Science - Special issue on selected papers of the International Workshop on Computing by Graph Transformation, Bordeaux, France, March 21–23, 1991
Discrete Applied Mathematics - Special issue on international workshop of graph-theoretic concepts in computer science WG'98 conference selected papers
Query evaluation via tree-decompositions
Journal of the ACM (JACM)
Algorithms for Guided Tree Automata
WIA '96 Revised Papers from the First International Workshop on Implementing Automata
Computing crossing numbers in quadratic time
Journal of Computer and System Sciences - STOC 2001
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Computing crossing number in linear time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Graph minors and parameterized algorithm design
The Multivariate Algorithmic Revolution and Beyond
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In 1990, Courcelle showed that every problem definable in Monadic Second-Order Logic (MSO) can be solved in linear time on graphs with bounded treewidth. This powerful and important theorem is amongst others the foundation for several fixed parameter tractability results. The standard proof of Courcelle's Theorem is to construct a finite bottom-up tree automaton that recognizes a tree decomposition of the graph. However, the size of the automaton, which is usually hidden as a constant in the Landau-notation, can become extremely large and cannot be bounded by any elemental function unless P=NP (Frick and Grohe, 2004). This makes the problem hard to tackle in practice, because it is just impossible to construct the tree automata. Aiming for a practical implementation, we give a proof of Courcelle's Theorem restricted to Extended MSO formulas of the form optU@?V@f(U), where @f is a first-order formula with vocabulary (adj, U) and opt@?{min,max}. Note that many optimization problems such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Independent Set can be expressed by such formulas. The proof uses a new technique based on using Hintikka game properties in dynamic programming. To demonstrate the usability of this approach, we present an implementation that solves such formulas on graphs with small pathwidth. It turns out that the large constants can be circumvented on graphs that are not too complex.