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Interval Representation of Balanced Separators in Graphs Avoiding a Minor

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Part of the Trends in Mathematics book series (RPCRMB,volume 14)

Abstract

We show that for any sufficiently large graph G avoiding \(K_k\) as a minor, we can map vertices \(v\in V(G)\) to intervals \(I(v)\subseteq [0,1]\) so that (1) \(I(u)\cap I(v)\ne \emptyset \) for each edge uv (2) the sum of the squares of the lengths of these intervals is \(O(k^6\log k)\), and (3) the average distance between the intervals is at least 1/25. Balanced separators of G of sublinear size (with various additional properties) can be read off this representation.

Keywords

  • Graph theory
  • Small separators
  • Minor-closed

Supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.

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Correspondence to Robert Šámal .

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Dvořák, Z., Pekárek, J., Šámal, R. (2021). Interval Representation of Balanced Separators in Graphs Avoiding a Minor. In: Nešetřil, J., Perarnau, G., Rué, J., Serra, O. (eds) Extended Abstracts EuroComb 2021. Trends in Mathematics(), vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83823-2_132

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