t-sails and sparse hereditary classes of unbounded tree-width
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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 × t)-wall and (4) a line graph of a subdivision of the (t × t)-wall. We are now able to add a further boundary object to this list, a subdivision of a 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 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.
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