Graph equivalences and decompositions definable in Monadic Second-Order Logic. The case of Circle Graphs

Many graph properties and graph transformations can be formalized inMonadic Second-Order logic. This language is the extension of First-Order logic allowing variables denoting sets of elements. In the case of graphs, these elements can be vertices, and in some cases edges. Monadic second-order graph properties can be checked in linear time on the class of graphs of tree-width at most k for any fixed k. These properties are Fixed Parameter Linear, for tree-width as a parameter. Monadic second-order logic as a language for specifying graph properties is interesting from several different points of view : we already mentioned complexity, but another point of view is that of Graph Grammars. For logicians, monadic second-order logic is attractive because relatively many classes of structures have a decidable theory for this language. In this communication we will discuss the point of view of Graph Theory. Many graph properties concerning colorings, forbidden configurations, connectivity are expressible in Monadic Second-Order logic, but also many graph theoretical constructions like the canonical decompositions of a graph in 2and 3-connected components, its modular and its split decompositions. We will review a number of cases where a set of graphs or of combinatorial objects is characterized by a common hierarchical decomposition. In the cases we will consider, the decomposition can be formalized in monadic second-order logic and from it, all graphs or objects of the corresponding set can be defined