Unbounded Tree-width in Sparse Hereditary Path-Star Graph Classes

Core Concepts

Sparse hereditary path-star graph classes defined by infinite words with an unbounded number of stars connecting to the path an unbounded number of times have unbounded tree-width, containing arbitrarily large subdivisions of t-sails as induced subgraphs.

Abstract

The content explores the tree-width properties of sparse hereditary graph classes, focusing on path-star graph classes defined by infinite words over a (possibly infinite) alphabet. Key highlights: It has long been known that the complete graph Kt, complete bipartite graph Kt,t, subdivision of the (t × t)-wall, and line graph of a subdivision of the (t × t)-wall are obstructions to bounded tree-width. The paper introduces a new obstruction - the subdivision of a t-sail graph. It is shown that a path-star class defined by an infinite word with an unbounded number of stars each connecting to the path an unbounded number of times has unbounded tree-width, containing arbitrarily large subdivisions of t-sails as induced subgraphs. These path-star classes are proven to be KKW-free (not containing the four basic obstructions) but not subclasses of circle graphs. A collection of nested words with a recursive structure is identified, where the corresponding path-star classes are KKW-free and contain a large t-sail subdivision if and only if they have large tree-width. It is shown that sparse hereditary graph classes of unbounded tree-width, including those defined by nested words, do not contain a minimal class of unbounded tree-width.

Stats

For arbitrarily large t, the complete graph Kt, complete bipartite graph Kt,t, subdivision of the (t × t)-wall, and line graph of a subdivision of the (t × t)-wall are obstructions to bounded tree-width.

Quotes

"It has long been known that the following basic objects are obstructions to bounded tree-width: for arbitrarily large t, (1) the complete graph Kt, (2) the complete bipartite graph Kt,t, (3) a subdivision of the (t × t)-wall and (4) the line graph of a subdivision of the (t × t)-wall." "We now add a further boundary object to this list, a subdivision of a t-sail."

Key Insights Distilled From

$t$-sails and sparse hereditary classes of unbounded tree-width

by Daniel Cocks at arxiv.org 04-30-2024

https://arxiv.org/pdf/2302.04783.pdf

Deeper Inquiries

How do the tree-width properties of path-star graph classes compare to those of other sparse hereditary graph classes, such as those with bounded degree or excluded minors

In the context of the paper, the tree-width properties of path-star graph classes differ from those of other sparse hereditary graph classes, such as those with bounded degree or excluded minors. Path-star graph classes are shown to have unbounded tree-width if they contain arbitrarily large subdivisions of a t-sail. This is in contrast to classes with bounded degree or excluded minors, where the tree-width is bounded if the class excludes a specific minor or has a bounded vertex degree. The paper demonstrates that path-star classes with an excluded minor or bounded vertex degree have unbounded tree-width only if they contain large subdivisions of a wall or the line graph of a subdivision of a wall. This distinction highlights the unique characteristics of path-star graph classes in relation to tree-width properties.

What other types of recursive structures, beyond nested words, could be used to define sparse hereditary graph classes with interesting tree-width characteristics

Beyond nested words, other types of recursive structures could be used to define sparse hereditary graph classes with interesting tree-width characteristics. One potential approach could involve using hierarchical structures, where graphs are defined based on nested levels of connectivity or hierarchy. For example, a graph class could be defined based on a recursive hierarchy of clusters or modules, where each level represents a different level of connectivity or interaction between components. By defining graph classes in this hierarchical manner, it may be possible to explore how different levels of connectivity impact the tree-width properties of the graphs. Another approach could involve using fractal structures to define graph classes. Fractals exhibit self-similarity at different scales, and this property could be leveraged to create graph classes with intricate recursive patterns. By defining graph classes based on fractal structures, it may be possible to study how the self-similar properties of fractals influence the tree-width characteristics of the graphs.

Are there any sparse hereditary graph classes that do contain a minimal class of unbounded tree-width, contrary to the conjecture presented in the paper

While the conjecture presented in the paper suggests that sparse hereditary graph classes of unbounded tree-width do not contain a minimal class of unbounded tree-width, there may be exceptions to this conjecture. It is possible that certain sparse hereditary graph classes exist that do contain a minimal class of unbounded tree-width. These classes would need to exhibit unique structural properties or characteristics that allow them to defy the general trend observed in the conjecture. Further research and analysis would be required to identify and study such exceptional cases and understand the underlying reasons for their deviation from the conjecture.