Neighborhood complexity of planar graphs
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Reidl, Sánchez Villaamil, and Stravopoulos (2019) characterized graph classes of bounded expansion as follows: A class 𝒞 closed under subgraphs has bounded expansion if and only if there exists a function f:ℕ→ℕ such that for every graph G ∈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 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.
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