Directed Acyclic Outerplanar Graphs Have Constant Stack Number

Core Concepts

Directed acyclic outerplanar graphs have bounded stack number, resolving a long-standing conjecture.

Abstract

The key insights and highlights of the content are:

The stack number of a directed acyclic graph (DAG) G is the minimum number of "stacks" required to partition the edges of G such that no two edges of the same stack cross each other in the topological ordering of the vertices.

The authors prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, resolving a conjecture by Heath, Pemmaraju and Trenk from 1999. This also implies that all upward outerplanar graphs have constant stack number.

As a complementary result, the authors construct a family of directed acyclic 2-trees that have unbounded stack number, refuting a conjecture by Nöllenburg and Pupyrev.

The authors introduce a novel technique called "directed H-partitions" as a key tool to prove the bounded stack number for outerplanar DAGs. This technique may be of independent interest.

The proof proceeds by partitioning the outerplanar DAG into "transitive parts" such that the contraction of each part into a single vertex yields a block-monotone DAG. The stack layouts of the individual blocks can then be combined into a single stack layout of the overall graph.

Stats

The content does not provide any specific numerical data or metrics to support the key arguments.

Quotes

The content does not contain any striking quotes that support the key logics.

Key Insights Distilled From

Directed Acyclic Outerplanar Graphs Have Constant Stack Number

by Paul Jungebl... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2211.04732.pdf

Deeper Inquiries

How can the directed H-partition technique be applied to other classes of directed graphs beyond outerplanar DAGs

The directed H-partition technique can be applied to other classes of directed graphs beyond outerplanar DAGs by leveraging its ability to partition a graph into subgraphs that can be efficiently stacked. This technique can be particularly useful for graphs with complex structures or specific constraints where finding an optimal stack layout is challenging. By identifying key partitions within the graph and applying the directed H-partition method, it becomes possible to simplify the stacking process and potentially reduce the stack number of the graph. This approach can be extended to various types of directed graphs, such as directed trees, directed acyclic graphs with specific properties, or directed graphs with hierarchical structures. The directed H-partition technique offers a systematic way to analyze and optimize stack layouts in a wide range of directed graph classes.

What are the implications of the unbounded stack number for directed acyclic 2-trees on the complexity of algorithms for finding optimal stack layouts

The unbounded stack number for directed acyclic 2-trees has significant implications for the complexity of algorithms aimed at finding optimal stack layouts. Since the stack number of a graph directly impacts the number of stacks required for a crossing-free layout, an unbounded stack number for directed acyclic 2-trees implies that the complexity of determining the optimal stack layout for such graphs is also unbounded. This poses a challenge for algorithm design and optimization, as the number of stacks needed to achieve a crossing-free layout may grow indefinitely with the size and structure of the graph. As a result, developing efficient algorithms for stack layout optimization for directed acyclic 2-trees becomes a more intricate task, requiring innovative approaches to handle the potentially unbounded stack number and ensure optimal layout solutions within reasonable computational bounds.

Is there a deeper connection between the stack number and the twist number of a directed graph, beyond the known polynomial relationship

The relationship between the stack number and the twist number of a directed graph goes beyond the known polynomial relationship and offers deeper insights into the graph's structural properties. While the twist number provides information about the crossing patterns of edges in a graph, the stack number determines the minimum number of stacks required for a crossing-free layout. Understanding how these two parameters interact can reveal underlying patterns in the graph's topology and connectivity. By exploring the connections between the stack number and the twist number, researchers can uncover more nuanced relationships between edge crossings and stack layouts, leading to enhanced graph visualization techniques, improved layout algorithms, and a deeper understanding of graph complexity. Investigating the interplay between the stack number and the twist number can shed light on the fundamental characteristics of directed graphs and their optimal layout representations.