Graph structure is a flexible concept covering many different
types of graph properties. Hierarchical decompositions yielding the
notions of tree-width and clique-width, expressed by terms written with
appropriate graph operations and associated with Monadic Second-order
Logic are important tools for the construction of Fixed-Parameter
Tractable algorithms and also for the extension of methods and results
of Formal Language Theory to the description of sets of finite graphs.
This informal overview presents the main definitions, results and open
problems and tries to answer some frequently asked questions.