[Paper Review] Low Polynomial Exclusion of Planar Graph Patterns

Jean‐Florent Raymond, Dimitrios M. Thilikos|arXiv (Cornell University)|May 30, 2013

Advanced Graph Theory Research19 references4 citations

TL;DR

This paper establishes low-degree polynomial upper bounds on treewidth for graphs excluding specific planar graph patterns as minors, including wheels, double wheels, pathwidth-2 graphs, and yurt graphs. It proves that treewidth is bounded by O(h), O(h² log²h), O(h²), and O(h⁴) respectively, significantly improving upon previous exponential bounds and advancing the understanding of polynomial grid exclusion in minor-closed graph families.

ABSTRACT

The celebrated grid exclusion theorem states that for every $h$-vertex planar graph $H$, there is a constant $c_{h}$ such that if a graph $G$ does not contain $H$ as a minor then $G$ has treewidth at most $c_{h}$. We are looking for patterns of $H$ where this bound can become a low degree polynomial. We provide such bounds for the following parameterized graphs: the wheel ($c_{h}=O(h)$), the double wheel ($c_{h}=O(h^2\cdot \log^{2} h)$), any graph of pathwidth at most 2 ($c_{h}=O(h^{2})$), and the yurt graph ($c_{h}=O(h^{4})$).

Motivation & Objective

• To determine whether the treewidth of graphs excluding a fixed planar minor H can be bounded by a low-degree polynomial in |V(H)|.
• To improve upon the previously known exponential or super-polynomial bounds in the grid exclusion theorem by identifying specific planar graph families with polynomial treewidth bounds.
• To establish tight or near-tight bounds for specific structured planar graphs such as wheels, double wheels, and yurt graphs.
• To investigate the optimality of these bounds by analyzing lower bounds using Ramanujan graphs and other extremal constructions.

Proposed method

• Using a combination of structural graph theory and minor containment arguments, the authors analyze the treewidth of graphs that exclude specific planar patterns as minors.
• Leveraging the fact that high treewidth forces the presence of certain substructures, the paper applies a variant of the Grid Minor Theorem to derive sufficient conditions for minor containment.
• The proof for the yurt graph relies on constructing a comb-like structure within high-treewidth graphs and using a subsequence argument to extract the required minor.
• For the double wheel and pathwidth-2 graphs, the authors use recursive decomposition and minor-closure properties to bound the treewidth in terms of the size of the excluded pattern.
• The analysis includes a detailed construction of the graph family Λ(C) for a comb C, which is used to simulate the target minor.
• Lower bounds are derived using Ramanujan graphs with high girth and large treewidth, showing that certain bounds are asymptotically tight.

Experimental results

Research questions

• RQ1 Can the treewidth of graphs excluding a planar minor H be bounded by a polynomial in |V(H)|, and if so, what is the best possible degree?
• RQ2 Is the treewidth bound for the wheel graph H of order h polynomial, and can it be improved to O(h)?
• RQ3 What is the best possible polynomial bound for the double wheel and yurt graph families, and are these bounds tight?
• RQ4 Are there structural classes of planar graphs for which the grid exclusion theorem yields a polynomial treewidth bound that is significantly better than the general case?
• RQ5 Can extremal graph constructions such as Ramanujan graphs be used to establish lower bounds on the best possible polynomial degree for exclusion of certain minor families?

Key findings

• The treewidth of any graph excluding a wheel of order h as a minor is bounded by O(h), which is tight up to constant factors.
• For the double wheel of order h, the treewidth is bounded by O(h² log²h), improving upon previous exponential bounds.
• Any graph of pathwidth at most 2, when excluded as a minor, yields a treewidth bound of O(h²) for the host graph.
• The yurt graph of order h is excluded as a minor only if the host graph has treewidth at most O(h⁴), which is the first such polynomial bound for this family.
• The paper provides a lower bound of Ω(h log h) for the treewidth bound when excluding certain minor families, suggesting that this degree may be optimal for some classes.
• The authors conjecture that the bound for double wheels is optimal at O(h), and that the Ω(h log h) lower bound for pathwidth-1 and yurt-like families is tight.