Embedding Graphs of Simple Treewidth into Sparse Products
Kevin Hendrey, David R. Wood, Jung Hon Yip

TL;DR
This paper investigates how graphs with bounded simple treewidth can be embedded into the undirected graph of a directed product of two graphs, revealing bounds, constructions, and limitations related to treewidth and indegree constraints.
Contribution
It establishes that graphs of simple treewidth k can be embedded into directed products of graphs with bounded indegree and treewidth, and proves the optimality of these bounds.
Findings
Graphs of simple treewidth k embed into directed products with bounded indegree and treewidth.
Outerplanar graphs embed into products of trees with maximum indegree 1.
Certain graphs of treewidth k cannot be embedded into products with lower indegree constraints.
Abstract
We study embeddings of graphs with bounded treewidth or bounded simple treewidth into the undirected graph underlying the directed product of two directed graphs. If the factors have bounded maximum indegrees, then the product graph has bounded maximum indegree and therefore is sparse. We prove that every graph of simple treewidth is contained in (ignoring edge directions) the directed product of directed graphs and , with and . Further, we show that this treewidth bound is best possible. Several corollaries follow from our results: every outerplanar graph is contained in the directed product of trees with maximum indegree , and every planar graph with treewidth is contained in a directed product of graphs with treewidth and maximum indegree . However, for…
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