Directed Acyclic Outerplanar Graphs Have Constant Stack Number

TL;DR

This work resolves a central question in directed linear layouts by proving that outerplanar DAGs have a constant stack number, achieved via the novel directed $H$-partition framework and a careful block decomposition that preserves small cut covers. The authors bound the stack number to $sn \leq 24776$ for outerplanar DAGs and show that upward outerplanar graphs inherit this property, while they also construct monotone directed acyclic $2$-trees with unbounded stack number to delineate the limits of the method. A key methodological contribution is the directed $H$-partition technique and its interplay with block-monotone quotients, enabling a transfer of known bounds on blocks to the whole graph. The results significantly advance the understanding of upward planarity in linear layouts and open several precise boundary questions for graphs of treewidth $2$ and beyond, with the directed $H$-partition framework likely to influence future work in directed graph layouts.

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 Nöllenburg and Pupyrev [GD 2023].

Figures (5)

• Figure 4: An outerplanar DAG $G$ with its block-cut tree. The construction sequence of $G$ is $1, 2, \ldots, 8$. In particular, the base edge is from $1$ to $2$. Dotted edges are non-edges of $G$ whose addition to $G$ gives an outerplanar $2$-tree.
• Figure 5: The four possible stackings of a new vertex $u$ onto a directed edge $vw$.
• Figure 6: A directed $H$-partition (orange) of a directed graph $G$ (left) and the quotient $H$ (right). The cut cover number of the part $P$ is $2$ as the two vertices marked in red cover all edges with exactly one endpoint in $P$.
• Figure 7: Left: A DAG $G$ with a directed $H$-partition as in \ref{['lem:H_partition_to_stack_layout']} with $p = 2$ and $t = 3$. Some edge directions are omitted for better readability. Middle: $H$ and its blocks. Right: The block-cut tree of $H$.
• Figure 9: The properties guaranteed by \ref{['lem:construct_H_partition']}. \ref{['fig:partition']} shows a larger example combing all properties. \ref{['prop:parts_maximal_transitive_subgraphs']} Each monotone vertex (thick) has its own part (orange) including all transitive vertices until the next monotone vertex, i.e., the transitive subgraph below it (right). \ref{['prop:two_paths']} Two paths $Q_1^+,Q_2^+$ cover all outer edges of part $P$. Adding another transitive vertex $z$ extends the path (dashed). Removing $v$ and $w$ yields $Q_1$ and $Q_2$. \ref{['prop:H_block_monotone']} Each block of the quotient (right) consists of a base edge (thick) and monotone vertices corresponding to the monotone vertices in $G$ (left). \ref{['prop:block_cut_cover_number_bounded']} Inside each block, the cut cover number is bounded, i.e., for each part $P$ there are four vertices (circled red) that cover all edges leaving $P$.

Theorems & Definitions (18)

• conjecture 1.1: Heath, Pemmaraju and Trenk, 1999 Heath1999_DAGs1
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Davies, 2022 Davies2022_ColoringCircleGraphs
• Theorem 3.1: Nöllenburg, Pupyrev Noellenburg2021_DAGsWithConstantStackNumber
• corollary 3.2
• Lemma 3.3
• corollary 3.4
• definition 3.5: Directed $H$-Partition, Cut Cover Number
• Lemma 3.6