On the Implicit Graph Conjecture

The implicit graph conjecture states that every sufficiently small, hereditary graph class has a labeling scheme with a polynomial-time computable label decoder. We approach this conjecture by investigating classes of label decoders defined in terms of complexity classes such as P and EXP. For instance, GP denotes the class of graph classes that have a labeling scheme with a polynomial-time computable label decoder. Until now it was not even known whether GP is a strict subset of GR. We show that this is indeed the case and reveal a strict hierarchy akin to classical complexity. We also show that classes such as GP can be characterized in terms of graph parameters. This could mean that certain algorithmic problems are feasible on every graph class in GP. Lastly, we define a more restrictive class of label decoders using first-order logic that already contains many natural graph classes such as forests and interval graphs. We give an alternative characterization of this class in terms of directed acyclic graphs. By showing that some small, hereditary graph class cannot be expressed with such label decoders a weaker form of the implicit graph conjecture could be disproven.Comment: 13 pages, MFCS 201

Shorter Labeling Schemes for Planar Graphs

An \emph{adjacency labeling scheme} for a given class of graphs is an algorithm that for every graph $G$ from the class, assigns bit strings (labels) to vertices of $G$ so that for any two vertices $u,v$, whether $u$ and $v$ are adjacent can be determined by a fixed procedure that examines only their labels. It is known that planar graphs with $n$ vertices admit a labeling scheme with labels of bit length $(2+o(1))\log{n}$. In this work we improve this bound by designing a labeling scheme with labels of bit length $(\frac{4}{3}+o(1))\log{n}$. In graph-theoretical terms, this implies an explicit construction of a graph on $n^{4/3+o(1)}$ vertices that contains all planar graphs on $n$ vertices as induced subgraphs, improving the previous best upper bound of $n^{2+o(1)}$. Our scheme generalizes to graphs of bounded Euler genus with the same label length up to a second-order term. All the labels of the input graph can be computed in polynomial time, while adjacency can be decided from the labels in constant time

Optimal induced universal graphs and adjacency labeling for trees

We show that there exists a graph $G$ with $O(n)$ nodes, where any forest of $n$ nodes is a node-induced subgraph of $G$. Furthermore, for constant arboricity $k$, the result implies the existence of a graph with $O(n^k)$ nodes that contains all $n$-node graphs as node-induced subgraphs, matching a $\Omega(n^k)$ lower bound. The lower bound and previously best upper bounds were presented in Alstrup and Rauhe (FOCS'02). Our upper bounds are obtained through a $\log_2 n +O(1)$ labeling scheme for adjacency queries in forests. We hereby solve an open problem being raised repeatedly over decades, e.g. in Kannan, Naor, Rudich (STOC 1988), Chung (J. of Graph Theory 1990), Fraigniaud and Korman (SODA 2010).Comment: A preliminary version of this paper appeared at FOCS'1