Monadic Second Order Logic on Graphs with Local Cardinality Constraints

Abstract

We show that all problems of the following form can be solved in polynomial time for graphs of bounded treewidth: Given a graph G and for each vertex v of G a set α(v) of non-negative integers. Is there a set S of vertices or edges of G such that S satisfies a fixed property expressible in monadic second order logic, and for each vertex v of G the number of vertices/edges in S adjacent/incident with v belongs to the set α(v)? A wide range of problems can be formulated in this way, for example Lovász’s General Factor Problem.