Monadic Second Order Logic on Graphs with Local Cardinality Constraints

We show that all problems of the following form can be solved in polynomial time for graphs of bounded treewidth: Given a graph Gand for each vertex vof Ga set i¾?(v) of non-negative integers. Is there a set Sof vertices or edges of Gsuch that Ssatisfies a fixed property expressible in monadic second order logic, and for each vertex vof Gthe number of vertices/edges in Sadjacent/incident with vbelongs to the set i¾?(v)? A wide range of problems can be formulated in this way, for example Lovasz's General Factor Problem.

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