  We introduce a new decomposition of a graphs into quasi-4-connected
components, where we call a graph quasi-4-connected if it is 3-connected and it
only has separations of order 3 that remove a single vertex. Moreover, we give
a cubic time algorithm computing the decomposition of a given graph.
  Our decomposition into quasi-4-connected components refines the well-known
decompositions of graphs into biconnected and triconnected components. We
relate our decomposition to Robertson and Seymour's theory of tangles by
establishing a correspondence between the quasi-4-connected components of a
graph and its tangles of order 4.
