  Motivated by the algorithmic study of 3-dimensional manifolds, we explore the
structural relationship between the JSJ decomposition of a given 3-manifold and
its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we
show that a "sufficiently complicated" JSJ decomposition of a 3-manifold
enforces a "complicated structure" for all of its triangulations. More
concretely, we show that, under certain conditions, the treewidth (resp.
pathwidth) of the graph that captures the incidences between the pieces of the
JSJ decomposition of an irreducible, closed, orientable 3-manifold M yields a
linear lower bound on its treewidth tw(M) (resp. pathwidth pw(M)), defined as
the smallest treewidth (resp. pathwidth) of the dual graph of any triangulation
of M.
  We present several applications of this result. We give the first example of
an infinite family of bounded-treewidth 3-manifolds with unbounded pathwidth.
We construct Haken 3-manifolds with arbitrarily large treewidth; previously the
existence of such 3-manifolds was only known in the non-Haken case. We also
show that the problem of providing a constant-factor approximation for the
treewidth (resp. pathwidth) of bounded-degree graphs efficiently reduces to
computing a constant-factor approximation for the treewidth (resp. pathwidth)
of 3-manifolds.
