Separating layered treewidth and row treewidth

Abstract

Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. 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.

An Improved Planar Graph Product Structure Theorem

“…We conclude with an open problem. Bose, Dujmović, Javarsineh, Morin, and Wood [7] defined the row treewidth of a graph G to be the minimum integer k such that G is isomorphic to a subgraph of H P for some graph H with treewidth k and for some path P . Theorem 1 by Dujmović et al [13] says that planar graphs have row treewidth at most 8.…”

The product structure of squaregraphs

“…The row treewidth of a graph $$G$$ is the minimum integer $$k$$ such that $$G\,⫇\,H\,\boxtimes \,P$$ for some graph $$H$$ with treewidth $$k$$ and path $$P$$ [8]. Theorem 2 says that every planar graph has row treewidth at most 6.…”

“…Also note that Theorem 1 cannot be strengthened by replacing “outerplanar graph” by “graph with bounded pathwidth”. Indeed, Bose, Dujmović, Javarsineh, Morin and Wood [8] showed that for every $$k\in {\mathbb{N}}$$ there is a tree $$T$$ (which is a squaregraph) such that for any graph $$H$$ and path $$P$$, if $$T\,⫇\,H\,\boxtimes \,P$$ then $$\text{pw}(H)\geqslant k$$.…”

“…Also note that Theorem 1 cannot be strengthened by replacing "outerplanar graph" by "graph with bounded pathwidth". Indeed, Bose, Dujmović, Javarsineh, Morin and Wood [8] showed that for every ∈ k there is a tree T (which is a squaregraph) such that for any graph H and path P, if…”