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.

翻译：长期以来已知以下基本结构是无界树宽的障碍：对于任意大的$t$，（1）完全图$K_t$的细分，（2）完全二分图$K_{t,t}$的细分，（3）$(t \times t)$-墙的细分，以及（4）$(t \times t)$-墙细分的线图。现我们能够在此列表中新增一个进一步的\emph{边界对象}，即\emph{$t$-帆}的细分。我们确定了不含上述四种基本障碍但包含任意大的$t$-帆或其细分的无界树宽继承图类。同时证明了这些稀疏图类不包含最小无界树宽类。这些结果通过研究\emph{路径-星}图类获得——这是一种由路径（或路径的并）与星林组合形成的稀疏继承图类，其特性由可能无限字母表上的无限词刻画。