  It has long been known that the following basic objects are obstructions to
bounded tree-width: for arbitrarily large $t$, $(1)$ the complete graph $K_t$,
$(2)$ the complete bipartite graph $K_{t,t}$, $(3)$ a subdivision of the $(t
\times t)$-wall and $(4)$ the line graph of a subdivision of the $(t \times
t)$-wall. We now add a further \emph{boundary object} to this list, a
\emph{$t$-sail}.
  These results have been obtained by studying sparse hereditary
\emph{path-star} graph classes, each of which consists of the finite induced
subgraphs of a single infinite graph whose edges can be partitioned into a path
(or forest of paths) with a forest of stars, characterised by an infinite word
over a possibly infinite alphabet. We show that a path-star class whose
infinite graph has an unbounded number of stars, each of which connects an
unbounded number of times to the path, has unbounded tree-width. In addition,
we show that such a class is not a subclass of the hereditary class of circle
graphs.
  We identify a collection of \emph{nested} words with a recursive structure
that exhibit interesting characteristics when used to define a path-star graph
class. These graph classes do not contain any of the four basic obstructions
but instead contain graphs that have large tree-width if and only if they
contain arbitrarily large $t$-sails. We show that these classes are infinitely
defined and, like classes of bounded degree or classes excluding a fixed minor,
do not contain a minimal class of unbounded tree-width.
