Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Delta, every graph with maximum degree at most Delta and sufficiently large treewidth contains either a subdivision of the (k xk)-wall or the line graph of a subdivision of the (k x k)-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows.