Definability Equals Recognizability of Partial 3-Trees and \sl k -Connected Partial \sl k -Trees

Abstract.

We consider graph decision problems on partial 3-trees that can be solved by a finite-state, leaf-to-root tree automaton processing a width-3 tree decomposition of the given graph. The class of yes-instances of such a problem is said to be recognizable by the tree automaton. In this paper we show that any such class of recognizable partial 3-trees is definable by a sentence in CMS logic. Here, CMS logic is the extension of Monadic Second-order logic with predicates to count the cardinality of sets modulo fixed integers. We also generalize this result to show that recognizability (by a tree automaton that processes width-k tree decompositions) implies CMS-definability for k -connected partial k -trees. The converse—definability implies recognizability—is known to hold for any class of partial k -trees, and even for any graph class with an appropriate definition of recognizability. It has been conjectured that recognizability implies definability over partial k -trees; but a proof was previously known only for k ≤ 2 . This paper proves the conjecture, and hence the equivalence of definability and recognizability, over partial 3-trees and k -connected partial k -trees. We use techniques that may lead to a proof of this equivalence over all partial k -trees.