Neighborhood complexity of planar graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909374758256640 |
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| author | Joret, Gwenaël Rambaud, Clément |
| author_facet | Joret, Gwenaël Rambaud, Clément |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12633 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Neighborhood complexity of planar graphs Joret, Gwenaël Rambaud, Clément Combinatorics Discrete Mathematics 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. |
| title | Neighborhood complexity of planar graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2302.12633 |