On the Implicit Graph Conjecture

TL;DR

This paper investigates the implicit graph conjecture by exploring classes of label decoders based on complexity classes, establishing a hierarchy, and characterizing classes via graph parameters and logic, which advances understanding of graph labeling schemes.

Contribution

It demonstrates a strict hierarchy within label decoder classes, characterizes some classes through graph parameters, and proposes a logical framework that challenges the conjecture's universality.

Findings

GP is a strict subset of GR, establishing a hierarchy.

Certain graph classes can be characterized by graph parameters.

A logical class of label decoders includes many natural graph classes.

Abstract

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.