The Definition in Monadic Second-Order Logic of Modular Decompositions of Ordered Graphs

Every graph can be represented uniquely in a hierarchical way by means of its modular decomposition. We establish that the modular decomposition of a linearly ordered graph is definable in monadic second-order (MS) logic in the graph itself. The modular decomposition does not depend on the linear order of the given graph. A set of graphs is recognizable w.r.t. the operations associated with graph substitution iff it is definable by a formula of an extension of MS logic based on the use of auxilliary linear orderings. This paper is an extended abstract: complete proofs can be found in [7].

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