Separating Layered Treewidth and Row Treewidth

Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. In particular, the layered treewidth of a graph G is the minimum integer k such that G has a tree-decomposition and a layering such that each bag has at most k vertices in each layer. The row treewidth of G is the minimum integer k such that G is isomorphic to a subgraph of H boxed times P for some graph H of treewidth at most k and for some path P. It follows from the definitions that the layered treewidth of a graph is at most its row treewidth plus 1. Moreover, a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, it has been open whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. In particular, for every integer k we describe a graph with layered treewidth 1 and row treewidth k. We also prove an analogous result for layered pathwidth and row pathwidth.