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On the model-checking of monadic second-order formulas with edge set quantifications. (English) Zbl 1236.05143
Summary: We extend clique-width to graphs with multiple edges. We obtain fixed-parameter tractable model-checking algorithms for certain monadic second-order graph properties that depend on the multiplicities of edges, with respect to this “new” clique-width. We define special tree-width, the variant of tree-width relative to tree-decompositions such that the boxes that contain a vertex are on a path originating from some fixed node. We study its main properties. This definition is motivated by the construction of finite automata associated with monadic second-order formulas using edge set quantifications. These automata yield fixed-parameter linear algorithms with respect to tree-width for the model-checking of these formulas. Their construction is much simpler for special tree-width than for tree-width, for reasons that we explain.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 03B15 Higher-order logic; type theory (MSC2010) 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C12 Distance in graphs 68Q60 Specification and verification (program logics, model checking, etc.)
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