Separating Geodesic Structure and Product Structure

The geodesic treewidth of a graph $G$ is the smallest $k$ for which there is a partition $\mathcal{P}$ into geodesics such that $G/\mathcal{P}$ has treewidth $k$, where $G/\mathcal{P}$ is obtained from $G$ by contracting each part of $\mathcal{P}$. Based on this notion, row treewidth was developed and is defined for a graph $G$ as the smallest $k$ such that $G \subseteq H \boxtimes P$ for some graph $H$ of treewidth $k$ and a path $P$. Equivalently, the row treewidth of a graph $G$ is the smallest $k$ for which there is a partition $\mathcal{P}$ into disjoint unions of geodesics that are aligned with respect to some layering such that $G/\mathcal{P}$ has treewidth $k$. We separate the two notions by showing that bounded row treewidth does not imply bounded geodesic treewidth and by presenting a polynomial-time algorithm to decide whether a graph of treewidth 2 has geodesic treewidth 1, which is known to be NP-hard for row treewidth [Biedl, Eppstein, Ueckerdt, 2025]. More generally, we provide an algorithm to decide whether a given graph has geodesic treewidth at most $d$ that is XP in the treewidth, whereas there is no such algorithm for row treewidth, unless P = NP [Biedl, Eppstein, Ueckerdt, 2025]. On the other hand, we show that computing the geodesic treewidth is NP-hard and that every graph with geodesic treewidth 1 has bounded row treewidth. Moreover, we improve the best known lower bound on the geodesic treewidth of planar graphs to 5.