Neighborhood complexity of planar graphs

Abstract

Reidl, Sánchez Villaamil, and Stravopoulos (2019) characterized graph classes of bounded expansion as follows: A class $\mathcal{C}$ closed under subgraphs has bounded expansion if and only if there exists a function $f:\mathbb{N} \to \mathbb{N}$ such that for every graph $G \in \mathcal{C}$, every nonempty subset $A$ of vertices in $G$ and every nonnegative integer $r$, the number of distinct intersections between $A$ and a ball of radius $r$ in $G$ is at most $f(r) |A|$. When $\mathcal{C}$ has bounded expansion, the function $f(r)$ coming from existing proofs is typically exponential. In the special case of planar graphs, it was conjectured by Sokołowski (2021) that $f(r)$ could be taken to be a polynomial. In this paper, we prove this conjecture: For every nonempty subset $A$ of vertices in a planar graph $G$ and every nonnegative integer $r$, the number of distinct intersections between $A$ and a ball of radius $r$ in $G$ is $O(r^4 |A|)$. We also show that a polynomial bound holds more generally for every proper minor-closed class of graphs.

Figures (5)

• Figure 1: The construction of $G'$ and $A'$ in the proof of Lemma \ref{['lemma:reduction_outerface']}
• Figure 2: The main step in the proof of Theorem \ref{['theorem:NC_planar_degree_6']}.
• Figure 3: Illustration of the proof of Lemma \ref{['lemma:structural_lemma_best_bound_planar']}.
• Figure 4: The construction of Theorem \ref{['theorem:construction_bounded_treewidth']} for $t=2$ and $\ell=5$. The dashed edges represent paths of length $r/2$, and the central tree is a subgraph of an $\ell \times \ell \times \ell$-grid where $\ell = \frac{r}{2(t+1)}$.
• Figure 5: The graph $G_\ell$ of Theorem \ref{['theorem:construction_1_planar']} for $\ell=3$. This graph is $(\ell-1)$-planar and has neighborhood complexity on $A=\{a_1,a_2,\dots, a_\ell\}$ at distance $\ell+1$ at least $2^\ell/\ell$.

Theorems & Definitions (58)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4: Corollary of Sauer–Shelah Lemma SAUER1972145shelah1972combinatorial
• proof
• Lemma 5
• proof
• Theorem 6: Bousquet and Thomassé bousquet_vc-dimension_2015
• Corollary 7
• Lemma 8