インサイト - Algorithms and Data Structures - # Graph Theory

Treewidth and Hadwiger Number Relationship in Hereditary Graph Classes

Q: Can the polynomial bound on the (tw, had)-bounding function be further improved for specific subclasses of (tw, had)-bounded graph classes?

It's highly plausible that the polynomial bound can be improved for specific subclasses of (tw, had)-bounded graph classes. Here's why: Theorem 2's reliance on general bounds: The O(had(G)⁹ polylog(had(G))) bound in Theorem 2 stems from using the current best result for the Grid Minor Theorem (Theorem 7). This theorem provides a general relationship between treewidth and grid minors, applicable to all graphs. However, when focusing on specific subclasses of (tw, had)-bounded graphs, this general bound might be overly pessimistic. Exploiting structural properties: (tw, had)-bounded classes, by definition, exhibit structural constraints that limit how "grid-like" they can become without forcing larger complete graph minors. Specific subclasses will have even stronger structural properties. For instance, outer-string graphs (Theorem 4) and their generalization to higher genus surfaces (Theorem 5) demonstrate linear (tw, had)-boundedness. This suggests that the relationship between treewidth, induced grid minors, and Hadwiger number might be tighter within these subclasses, allowing for improved bounds. Potential for future research: Identifying sharper relationships between treewidth and grid minors within specific (tw, had)-bounded classes is an open avenue for research. This could involve: Refined Grid Minor Theorems: Proving specialized versions of the Grid Minor Theorem tailored to the specific structural properties of the subclass. Alternative approaches: Exploring techniques beyond grid minors to directly relate treewidth and Hadwiger number within the subclass. In summary, while Theorem 2 provides a general polynomial bound, there's a strong possibility for achieving better bounds by leveraging the specific structural characteristics of subclasses within (tw, had)-bounded graph classes.

Q: Could there be a counterexample to the linear (tw, had)-bounding conjecture in a (tw, had)-bounded class with a complex, non-planar obstruction?

It's certainly conceivable that a counterexample to the linear (tw, had)-bounding conjecture could exist within a (tw, had)-bounded class characterized by a complex, non-planar obstruction. Here's why this possibility should be considered: Limited understanding of complex obstructions: While excluding a planar graph as an induced minor guarantees (tw, had)-boundedness, our understanding of the structural implications of excluding complex, non-planar graphs is still developing. These complex obstructions might allow for intricate graph constructions that exhibit a non-linear relationship between treewidth and Hadwiger number. Subtle interplay of parameters: The interplay between treewidth, Hadwiger number, and the excluded induced minor can be quite subtle. It's possible that certain complex obstructions impose constraints that, while preventing arbitrarily large planar induced minors, still permit treewidth to grow super-linearly with respect to the Hadwiger number. Need for novel constructions: Finding such a counterexample would likely require constructing a family of graphs within the class where: The Hadwiger number grows relatively slowly. The treewidth increases at a faster, non-linear rate, despite the excluded induced minor. In essence, the possibility of a counterexample highlights the potential complexity hidden within (tw, had)-bounded classes defined by non-planar obstructions. Further research is needed to either strengthen the conjecture by ruling out such counterexamples or to discover families of graphs that exhibit a non-linear relationship.

核心概念

A hereditary graph class exhibits a bounded relationship between treewidth and Hadwiger number if and only if it excludes a planar graph as an induced minor. This relationship can be bounded by a polynomial function, and it is conjectured to be linear.

要約

Bibliographic Information: Campbell, R., Davies, J., Distel, M., Frederickson, B., Gollin, J. P., Hendrey, K., Hickingbotham, R., Wiederrecht, S., Wood, D. R., & Yepremyan, L. (2024, October 25). Treewidth, Hadwiger Number, and Induced Minors. [Preprint]. arXiv:2410.19295v1
Research Objective: This paper investigates the relationship between treewidth and Hadwiger number in hereditary graph classes, aiming to characterize classes where large treewidth implies the existence of a large complete graph minor.
Methodology: The authors employ techniques from structural graph theory, particularly focusing on induced minors, vertex-minors, and separators, to analyze the properties of graphs with specific treewidth and Hadwiger number constraints.
Key Findings: The paper presents several key findings:
- A hereditary graph class has a bounded relationship between treewidth and Hadwiger number (termed (tw, had)-bounded) if and only if it excludes some planar graph as an induced minor.
- Every (tw, had)-bounded graph class has a (tw, had)-bounding function that is at most polynomial in the Hadwiger number.
- The paper verifies the conjecture of a linear (tw, had)-bounding function for specific graph classes like outer-string graphs, (g, c)-outer-string graphs, and low-rank perturbations of circle graphs.
Main Conclusions: The authors conclude that the existence of a bounded relationship between treewidth and Hadwiger number in a hereditary graph class is entirely determined by the exclusion of a planar graph as an induced minor. They provide a polynomial bound for this relationship and conjecture it to be linear, supporting this conjecture for several important graph classes.
Significance: This research significantly contributes to the field of structural graph theory by providing a deeper understanding of the interaction between treewidth and Hadwiger number. The characterization of (tw, had)-bounded graph classes and the conjecture regarding linear (tw, had)-bounding functions open new avenues for research in graph minor theory and algorithmic graph theory.
Limitations and Future Research: While the paper confirms the linear (tw, had)-bounding function for specific graph classes, proving the conjecture for all (tw, had)-bounded classes remains an open problem. Further research could explore tighter bounds for the (tw, had)-bounding function and investigate the implications of these findings for algorithmic problems on graphs with excluded planar induced minors.

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統計

The best known bound for the Grid Minor Theorem implies a (tw, had)-bounding function in O(t^26 + t^24k).
The paper improves the bound for Theorem 1 to f(k, t) ∈ O(kt).
For any graph H, there is an integer cH such that every H-minor-free graph has treewidth at most cH if and only if H is planar.
Planar graphs have Hadwiger number at most 4.
Kt-minor-free circle graphs have treewidth less than 12t.
The (k × k)-grid is a minor of Kk^2.
Kk^2−1 has treewidth k^2 − 2 and contains no (k × k)-grid minor.

引用

"Treewidth and Hadwiger number are two of the most important parameters in structural graph theory."
"This paper studies graph classes in which large treewidth implies the existence of a large complete graph minor."
"We characterise (tw, had)-bounded graph classes as those that exclude some planar graph as an induced minor."
"More strongly, we conjecture that every (tw, had)-bounded hereditary graph class is linearly (tw, had)-bounded."

抽出されたキーインサイト

Treewidth, Hadwiger Number, and Induced Minors

by Rutger Campb... 場所 arxiv.org 10-28-2024

https://arxiv.org/pdf/2410.19295.pdf

Treewidth, Hadwiger Number, and Induced Minors

深掘り質問

Can the polynomial bound on the (tw, had)-bounding function be further improved for specific subclasses of (tw, had)-bounded graph classes?

It's highly plausible that the polynomial bound can be improved for specific subclasses of (tw, had)-bounded graph classes. Here's why:

Theorem 2's reliance on general bounds: The O(had(G)⁹ polylog(had(G))) bound in Theorem 2 stems from using the current best result for the Grid Minor Theorem (Theorem 7). This theorem provides a general relationship between treewidth and grid minors, applicable to all graphs. However, when focusing on specific subclasses of (tw, had)-bounded graphs, this general bound might be overly pessimistic.
Exploiting structural properties:  (tw, had)-bounded classes, by definition, exhibit structural constraints that limit how "grid-like" they can become without forcing larger complete graph minors.  Specific subclasses will have even stronger structural properties. For instance, outer-string graphs (Theorem 4) and their generalization to higher genus surfaces (Theorem 5) demonstrate linear (tw, had)-boundedness. This suggests that the relationship between treewidth, induced grid minors, and Hadwiger number might be tighter within these subclasses, allowing for improved bounds.
Potential for future research:  Identifying sharper relationships between treewidth and grid minors within specific (tw, had)-bounded classes is an open avenue for research. This could involve:

Refined Grid Minor Theorems:  Proving specialized versions of the Grid Minor Theorem tailored to the specific structural properties of the subclass.
Alternative approaches: Exploring techniques beyond grid minors to directly relate treewidth and Hadwiger number within the subclass.
In summary, while Theorem 2 provides a general polynomial bound, there's a strong possibility for achieving better bounds by leveraging the specific structural characteristics of subclasses within (tw, had)-bounded graph classes.

Could there be a counterexample to the linear (tw, had)-bounding conjecture in a (tw, had)-bounded class with a complex, non-planar obstruction?

It's certainly conceivable that a counterexample to the linear (tw, had)-bounding conjecture could exist within a (tw, had)-bounded class characterized by a complex, non-planar obstruction. Here's why this possibility should be considered:

Limited understanding of complex obstructions: While excluding a planar graph as an induced minor guarantees (tw, had)-boundedness, our understanding of the structural implications of excluding complex, non-planar graphs is still developing. These complex obstructions might allow for intricate graph constructions that exhibit a non-linear relationship between treewidth and Hadwiger number.
Subtle interplay of parameters: The interplay between treewidth, Hadwiger number, and the excluded induced minor can be quite subtle. It's possible that certain complex obstructions impose constraints that, while preventing arbitrarily large planar induced minors, still permit treewidth to grow super-linearly with respect to the Hadwiger number.
Need for novel constructions:  Finding such a counterexample would likely require constructing a family of graphs within the class where:

The Hadwiger number grows relatively slowly.
The treewidth increases at a faster, non-linear rate, despite the excluded induced minor.
In essence, the possibility of a counterexample highlights the potential complexity hidden within (tw, had)-bounded classes defined by non-planar obstructions. Further research is needed to either strengthen the conjecture by ruling out such counterexamples or to discover families of graphs that exhibit a non-linear relationship.

How does the relationship between treewidth and Hadwiger number influence the computational complexity of finding specific substructures within these graph classes?

The relationship between treewidth and Hadwiger number has profound implications for the computational complexity of finding substructures within (tw, had)-bounded graph classes.
Positive Influence:

Bounded treewidth enables efficient algorithms: Many NP-hard problems become polynomial-time solvable on graphs of bounded treewidth. This is because tree decompositions allow for dynamic programming algorithms that exploit the tree-like structure.  Therefore, in (tw, had)-bounded classes, if we can determine that the Hadwiger number is small, we automatically obtain a bound on the treewidth, making a variety of subgraph finding problems tractable.
Examples:

Finding cliques:  The maximum clique problem, which is NP-hard in general, can be solved in polynomial time on graphs of bounded treewidth.
Finding induced paths or cycles:  Finding long induced paths or cycles, also NP-hard generally, becomes easier with bounded treewidth.
Challenges with unbounded Hadwiger number:

Unbounded treewidth poses difficulties: When the Hadwiger number is unbounded, the treewidth can also be unbounded within a (tw, had)-bounded class. This often makes subgraph finding problems NP-hard.
Example: Even within planar graphs (Hadwiger number at most 4), finding a longest induced path remains NP-hard.
The role of the (tw, had)-bounding function:

Polynomial bounds offer limited help: If a (tw, had)-bounded class has a polynomial (tw, had)-bounding function, it provides a polynomial-time reduction to a bounded-treewidth instance. However, the exponent in the polynomial bound can significantly impact practical efficiency.
Linear bounds are highly desirable: Linear (tw, had)-boundedness is particularly valuable. It implies that a small Hadwiger number directly translates to a small treewidth, making many subgraph finding problems practically solvable.
In conclusion: The relationship between treewidth and Hadwiger number is crucial for understanding the complexity of finding substructures. (tw, had)-boundedness offers a pathway to efficient algorithms, especially when the bounding function is linear. However, unbounded Hadwiger number often implies unbounded treewidth, posing significant computational challenges even within these restricted graph classes.