A Monadic Second-Order Definition of the Structure of Convex Hypergraphs

We consider hypergraphs with unordered hyperedges of unbounded rank. Each hypergraph has a unique modular decomposition, which is a tree, the nodes of which consist of certain prime subhypergraphs of the considered hypergraph. One can define this decomposition by Monadic Second-order (MS) logical formulas. Such a hypergraph is convex if the vertices are linearly ordered in such a way that the hyperedges form intervals. Our main result says that the unique linear order witnessing the convexity of a prime hypergraph can be defined in MS logic. Using other results, we deduce that, for any set of bipartite graphs corresponding in the usual way to convex hypergraphs, if it has a decidable monadic theory, then it has bounded clique-width. This yields a new case of validity of a conjecture which is still open.