Recognizability Equals Definability for Graphs of Bounded Treewidth and Bounded Chordality

The technique of translating between monadic second-order logic (MSOL) formulae and equivalent automata has a long history. An early result is a theorem of Büchi from 1960 [2] showing that the languages accepted by finite automata are exactly the MSOL-definable sets of strings. Viewed as a result on families of graphs this can be seen as establishing that recognizability equals definability for labeled paths. In a series of seminal papers starting in 1990 Courcelle [3] introduced the concept of a recognizable set of graphs and began an investigation of the monadic second-order logic of graphs and of sets of graphs. He established that any MSOL-definable family of graphs is recognizable, but showed that for graphs in general the converse cannot hold. However, for unordered unbounded trees Courcelle [3] did establish that recognizability equals definability, using a counting monadic second-order logic (CMSOL). The following quote from a later paper in the series illustrates the situation at the time: It is not clear at all how an automaton should traverse a graph. A ”general” graph has no evident structure, whereas a word or a tree is (roughly speaking) its own algebraic structure. [4] The proposal for how to deal with this, using tree-decompositions of graphs, has nowadays become standard, see the recent book of Courcelle and Engelfriet [6], and will be the main tool that we use also in this paper. Courcelle proceeded to show that recognizability equals CMSOL-definability for graphs of treewidth at most two and conjectured that recognizability equals CMSOL-definability for graphs of bounded treewidth [4]. In this paper we establish that recognizability equals CMSOL-definability for graphs of bounded treewidth and bounded chordality (no chordless cycles of length larger than a constant c), thereby proving a special case of Courcelle’s conjecture. Let us mention related work on the conjecture. In 1995 Kaller [10] established the special case of graphs of treewidth at most 3 and k-connected graphs of treewidth at most k. Two conference papers from 1997 [9] and 1998 [11] claimed to be able to prove the conjecture for, respectively, graphs of bounded pathwidth and graphs of bounded treewidth. However, full versions have not appeared of any of these two papers, and people in the field do not consider either of them satisfactory. Very recently, Jaffke and Bodlaender showed that recognizability implies MSOL-definability for Halin graphs, which are of treewidth 3, and for some related graph classes [8]. The main difficulty in proving Courcelle’s conjecture for a class of graphs is to define in CMSOL, for any graph in the class, concrete minimum-width tree-decompositions, and that is what we do in this paper. We end the paper by arguing that to prove the full conjecture an extension of the techniques used in this paper will be needed.