The modular decomposition of countable graphs. Definition and construction in monadic second-order logic

We consider the notion of modular decomposition for countable graphs. The modular decomposition of a graph given with an enumeration of its set of vertices can be defined by formulas of monadic second-order logic. Another result is the definition of a representation of modular decompositions by low degree relational structures. Such relational structures can also be defined in the considered graph by monadic second-order formulas.