$t$-sails and sparse hereditary classes of unbounded tree-width

It has long been known that the following basic objects are obstructions to bounded tree-width: for arbitrarily large $t$, $(1)$ a subdivision of the complete graph $K_t$, $(2)$ a subdivision of the complete bipartite graph $K_{t,t}$, $(3)$ a subdivision of the $(t \times t)$-wall and $(4)$ a line graph of a subdivision of the $(t \times t)$-wall. We are now able to add a further \emph{boundary object} to this list, a subdivision of a \emph{$t$-sail}. We identify hereditary graph classes of unbounded tree-width that do not contain any of the four basic obstructions but instead contain arbitrarily large $t$-sails or subdivisions of a $t$-sail. We also show that these sparse graph classes do not contain a minimal class of unbounded tree-width. These results have been obtained by studying \emph{path-star} graph classes, a type of sparse hereditary graph class formed by combining a path (or union of paths) with a forest of stars, characterised by an infinite word over a possibly infinite alphabet.