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 $t$, $\left(1\right)$ the complete graph $K_t$, $\left(2\right)$ the complete bipartite graph $K_{t,t}$, $\left(3\right)$ a subdivision of the $(t\times t)$-wall and $\left(4\right)$ the line graph of a subdivision of the $(t\times t)$-wall. We now add a further boundary object to this list, a $t$-sail. These results have been obtained by studying sparse hereditary 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 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.