[Paper Review] Product structure of graph classes with strongly sublinear separators

David R. Wood, Wood, David R.|arXiv (Cornell University)|Aug 22, 2022

Advanced Graph Theory Research4 citations

TL;DR

This paper establishes that hereditary graph classes with strongly sublinear separators—where separators grow as $O(n^{1- ho})$ for some $\rho > 0$—admit a product structure as subgraphs of the strong product of a bounded tree-depth graph and a complete graph of size $O(n^{1-\rho+\delta})$, for any $\delta > 0$. The key contribution is a tight characterization showing that the dependence on $\delta$ and the $O(\log \log n)$ tree-depth bound are optimal, resolving a central open problem in product structure theory.

ABSTRACT

We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class $\mathcal{G}$ admits $O(n^{1-ε})$ separators, then for any fixed $δ\in(0,ε)$ every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-depth and a complete graph of size $O(n^{1-ε+δ})$. This result holds with $δ=0$ if we allow $H$ to have tree-depth $O(\log\log n)$. Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on $δ$ in our results and the above $ ext{td}(H)\in O(\log\log n)$ bound are both best possible. We prove that $n$-vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth $t$ and a complete graph of size $O(n^{1/t})$, which is best possible. Finally, we investigate the conjecture that for any hereditary graph class $\mathcal{G}$ that admits $O(n^{1-ε})$ separators, every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-width and a complete graph of size $O(n^{1-ε})$. We prove this for various classes $\mathcal{G}$ of interest.

Motivation & Objective

• To characterize hereditary graph classes with strongly sublinear separators using product structure theorems.
• To determine the optimal dependence on the separator exponent $\epsilon$ in the size of the complete graph factor in strong product decompositions.
• To investigate whether every $n$-vertex graph in such a class is a subgraph of $H \boxtimes K_m$ with $H$ of bounded treewidth and $m = O(n^{1-\epsilon})$, as conjectured.
• To prove optimality of the bounds on tree-depth and the $\delta$-dependence in the exponent.

Proposed method

• Use of strong product decompositions to represent graphs in hereditary classes with strongly sublinear separators.
• Application of isoperimetric inequalities on grid graphs to derive lower bounds on the required size of the complete graph factor.
• Construction of extremal graphs via iterated subdivisions of trees to demonstrate tightness of bounds.
• Use of tree-depth and path-decomposition techniques to bound structural parameters in the decomposition.
• Proof by contradiction to show that the $\delta$-dependence and $O(\log \log n)$ tree-depth bounds are optimal.
• Leverage known results on minor-closed and geometric graph classes to verify the conjecture in special cases.

Experimental results

Research questions

• RQ1 Can every hereditary graph class with $O(n^{1-\epsilon})$ separators be represented as a subgraph of $H \boxtimes K_m$ with $H$ of bounded tree-depth and $m = O(n^{1-\epsilon+\delta})$ for any $\delta > 0$?
• RQ2 Is the $O(\log \log n)$ tree-depth bound for $H$ tight when $\delta = 0$?
• RQ3 Can the dependence on $\delta$ in the exponent be eliminated, i.e., is $m = O(n^{1-\epsilon})$ achievable with $H$ of bounded tree-width?
• RQ4 Are the bounds on tree-depth and $\delta$-dependence optimal for such product decompositions?
• RQ5 Does the conjecture hold for specific classes like touching graphs of spheres, $k$-crossing-degenerate graphs, or string graphs?

Key findings

• Every $n$-vertex graph in a hereditary graph class with $O(n^{1-\epsilon})$ separators is a subgraph of the strong product of a graph $H$ with tree-depth $O(\log \log n)$ and a complete graph of size $O(n^{1-\epsilon+\delta})$ for any $\delta > 0$.
• The dependence on $\delta$ in the exponent is optimal, as shown by constructing extremal graphs where reducing $\delta$ forces a larger complete graph factor.
• The $O(\log \log n)$ tree-depth bound for $H$ when $\delta = 0$ is optimal, as demonstrated via isoperimetric arguments on grid-like graphs.
• For graphs of bounded treewidth $t$, the paper proves a tight product structure: they are subgraphs of $H \boxtimes K_m$ with $H$ of tree-depth $t$ and $m = O(n^{1/t})$, which is optimal.
• The conjecture that $m = O(n^{1-\epsilon})$ with $H$ of bounded tree-width holds for several important classes, including planar graphs, graphs of bounded Euler genus, and minor-closed classes.
• The paper shows that the conjecture fails if $\mathcal{G}$ is not hereditary, by constructing a single graph with unbounded complete subgraphs that cannot be embedded in such a product with bounded tree-width $H$.