Definability Equals Recognizability for k-Outerplanar Graphs

Abstract

One of the most famous algorithmic meta-theorems states that every graph property that can \nbe defined by a sentence in counting monadic second order logic (CMSOL) can be checked in \nlinear time for graphs of bounded treewidth, which is known as Courcelle\xe2\x80\x99s Theorem [6]. These \nalgorithms are constructed as finite state tree automata, and hence every CMSOL-definable \ngraph property is recognizable. Courcelle also conjectured that the converse holds, i.e. every \nrecognizable graph property is definable in CMSOL for graphs of bounded treewidth. We prove \nthis conjecture for k-outerplanar graphs, which are known to have treewidth at most 3k \xe2\x88\x92 1 [2]