Optimal labelling schemes for adjacency, comparability, and reachability

@article{Bonamy2021OptimalLS,
  title={Optimal labelling schemes for adjacency, comparability, and reachability},
  author={Marthe Bonamy and Louis Esperet and Carla Groenland and Alex D. Scott},
  journal={Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing},
  year={2021}
}
  • Marthe Bonamy, Louis Esperet, A. Scott
  • Published 3 December 2020
  • Computer Science, Mathematics
  • Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing 2Ω(n2) n-vertex graphs as n→ ∞. This regime contains many classes of interest, for instance perfect graphs or comparability graphs, for which we obtain an adjacency labelling scheme with labels of n/4+o(n) bits per vertex. This implies the existence of a reachability labelling scheme for digraphs with labels of n/4+o(n) bits per vertex and comparability labelling scheme for posets with labels… 
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4 Citations
Background Citations
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4 Citations

The Implicit Graph Conjecture is False
TLDR
The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at most factorial speed of growth admits an efficient implicit representation, and this paper proves this conjecture by establishing the existence of hereditary graph families with factorialSpeed of growth that require codes of length n.
Implicit representation of sparse hereditary families
TLDR
It is proved that for every ε > 0 there is an integer d ≥ 1 so that if F is a hereditary family with speed f(n) ≤ 2(1/4−ε)n2 then Fn admits an implicit representation of size O(n logn).
Isometric Universal Graphs
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It is shown that for any integer n > 1, there is a graph on 3 2 n) vertices that contains all n-vertex graphs as isometric subgraphs.
Boolean dimension and dim-boundedness: Planar cover graph with a zero
TLDR
The Boolean dimension result is proved in tandem with a second result showing that the dimension of a poset with a planar cover graph and a unique minimal element is bounded by a linear function of its standard example number.

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