Definability equals recognizability for graphs of bounded treewidth
  We prove a conjecture of Courcelle, which states that a graph property is definable in MSO with modular counting predicates on graphs of constant treewidth if, and only if it is recognizable in the following sense: constant-width tree decompositions of graphs satisfying the property can be recognized by tree automata. While the forward implication is a classic fact known as Courcelle's theorem, the converse direction remained open 
Miko{\l}aj Boja\'nczyk and Micha{\l} Pilipczuk
cs.LO cs.CC cs.DM cs.DS cs.FL