Graph structure and Monadic second-order logic
نویسنده
چکیده
Exclusion of minor, vertex-minor, induced subgraph Tree-structuring Monadic second-order logic : expression of properties, queries, optimization functions, number of configurations. Mainly useful for tree-structured graphs (Second-order logic useless) Tools to be presented Algebraic setting for tree-structuring of graphs Recognizability = finite congruence ≡ inductive computability ≡ finite deterministic automaton on terms Fefermann-Vaught : MS definability ⇒ recognizability. 3 History : Confluence of 4 independent research directions, now intimately related : 1. Polynomial algorithms for NP-complete and other hard problems on particular classes of graphs, and especially hierarchically structured ones : series-parallel graphs, cographs, partial k-trees, graphs or hypergraphs of tree-width < k, graphs of clique-width < k. 2. Excluded minors and related notions of forbidden configurations (matroid minors, « vertex-minors »). 3. Decidability of Monadic Second-Order logic on classes of finite graphs, and on infinite graphs. 4. Extension to graphs and hypergraphs of the main concepts of Formal
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Graph equivalences and decompositions definable in Monadic Second-Order Logic. The case of Circle Graphs
Many graph properties and graph transformations can be formalized inMonadic Second-Order logic. This language is the extension of First-Order logic allowing variables denoting sets of elements. In the case of graphs, these elements can be vertices, and in some cases edges. Monadic second-order graph properties can be checked in linear time on the class of graphs of tree-width at most k for any ...
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