Directed Acyclic Outerplanar Graphs Have Constant Stack Number

TL;DR

This paper proves that directed acyclic outerplanar graphs have a constant stack number, confirming a longstanding conjecture and advancing understanding of upward outerplanar graphs, while also introducing a new technique of directed H-partitions.

Contribution

It establishes a constant bound on the stack number for directed acyclic outerplanar graphs and introduces directed H-partitions as a novel analytical tool.

Findings

Directed acyclic outerplanar graphs have bounded stack number.

All upward outerplanar graphs have constant stack number.

Constructed directed acyclic 2-trees with unbounded stack number.

Abstract

The stack number of a directed acyclic graph $G$ is the minimum $k$ for which there is a topological ordering of $G$ and a $k$-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, which gives a positive answer to a conjecture by Heath, Pemmaraju and Trenk [SIAM J. Computing, 1999]. As an immediate consequence, this shows that all upward outerplanar graphs have constant stack number, answering a question by Bhore et al. [Eur. J. Comb., 2023] and thereby making significant progress towards the problem for general upward planar graphs originating from Nowakowski and Parker [Order, 1989]. As our main tool we develop the novel technique of directed $H$-partitions, which might be of independent interest. We complement the bounded stack number for directed acyclic outerplanar graphs by constructing a family of directed acyclic 2-trees that have unbounded stack number, thereby refuting a conjecture by N\"ollenburg and Pupyrev [GD 2023].