Fly-automata for checking monadic second-order properties of graphs of bounded tree-width

Every graph property expressible in monadic second-order (MSO) logic, possibly with quantifications over edges, can be checked in linear time on graphs of bounded tree-width, in particular by means of finite automata running on terms denoting tree-decompositions. However, implementing these automata is difficult because of their huge sizes. Fly-automata (FA) are deterministic automata that compute the necessary states and transitions when running (instead of looking into tables); they allow us to overcome this difficulty. In previous works, we constructed FA to check MSO properties of graphs of bounded clique-width. An MSO property with edge quantifications (called an MSO2 property) of a graph is an MSO property of its incidence graph and, on the other hand, graphs of tree-width k have incidence graphs of clique-width O(k). Thus, our existing constructions can be used for MSO2 properties of graphs of bounded tree-width. We examine concrete aspects of this adaptation.