Circle graphs and monadic second-order logic

A circle graph is the intersection graph of a set of chords of a circle. If a circle graph is prime for the split (or join) decomposition defined by Cunnigham, it has a unique representation as a set of intersecting chords, and we prove that this representation can be defined by monadic second-order formulas. By using the (canonical) split decomposition of a circle graph, one can define in monadic second-order logic all its chord representations formalized as words with two occurrences of each letter. This construction uses the general result that the split decomposition of a graph can be constructed in monadic second-order logic. As a consequence we prove that a set of circle graphs has bounded clique-width if and only if all their chord diagrams have bounded tree-width. We also prove that the order of first occurrences of the letters in a double occurrence word w representing a given connected circle graph determines this word w in a unique way.