# 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} }

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…

## 4 Citations

The Implicit Graph Conjecture is False

- MathematicsArXiv
- 2021

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

- MathematicsArXiv
- 2022

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

- MathematicsSIAM J. Discret. Math.
- 2021

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

- Mathematics, Computer ScienceArXiv
- 2022

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.

## References

SHOWING 1-10 OF 37 REFERENCES

Efficient Labeling for Reachability in Digraphs

- Computer ScienceArXiv
- 2020

This work considers labeling nodes of a directed graph for reachability queries and designs a scheme with labels consisting of $n/3+o(n)$ bits, which is optimal, up to the lower order term.

Adjacency Labeling Schemes and Induced-Universal Graphs

- Mathematics, Computer ScienceSTOC
- 2015

An induced-universal graph for n-vertex graphs containing only O(2n/2) vertices is obtained, which is optimal up to a multiplicative constant, solving an open problem of Vizing from 1968.

Compact oracles for reachability and approximate distances in planar digraphs

- Computer Science, MathematicsJACM
- 2004

The technique is based on a novel dipath decomposition of planar digraphs that instead of using the standard separator with O(/spl radic/n) vertices, in effect finds a separator using a constant number of dipaths.

The Penultimate Rate of Growth for Graph Properties

- MathematicsEur. J. Comb.
- 2001

The behavior of properties with speeds between these latter bounds are discussed, i.e., between n(1+o(1))n and 2(1/2+o (1)n2/2.

Asymptotically optimal induced universal graphs

- Mathematics
- 2017

We prove that the minimum number of vertices of a graph that contains every graph on k vertices as an induced subgraph is $${(1+o(1)) 2^{(k-1)/2}}$$(1+o(1))2(k-1)/2. This improves earlier estimates…

Succinct Posets

- Computer Science, MathematicsAlgorithmica
- 2015

A succinct data structure for representing a poset that, given two elements, can report whether one precedes the other in constant time is designed, and a slight extension to this data structure yields a succinct oracle for reachability in arbitrary directed graphs.

Colorings and orientations of graphs

- MathematicsComb.
- 1992

Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques by proving that there is a legal vertex-coloring of G assigning to each vertexv a color fromS(v).

Implicit representations and factorial properties of graphs

- Mathematics, Computer ScienceDiscret. Math.
- 2015

AN EXTREMAL PROBLEM IN GRAPH THEORY

- Mathematics
- 2001

G(?z; I) will denote a graph of n vertices and 1 edges. Let fO(lz, K) be the smallest integer such that there is a G (n; f,, (n, k)) in which for every set of K vertices there is a vertex joined to…