Local Certification of Some Geometric Intersection Graph Classes

TL;DR

This paper develops compact proof labeling schemes with logarithmic certificates for recognizing various geometric intersection graph classes in distributed systems, and establishes tight lower bounds for these certificates.

Contribution

It introduces the first logarithmic-sized certificates for local recognition of several geometric intersection graph classes and proves tight lower bounds for their certificate sizes.

Findings

Logarithmic-sized certificates for interval, chordal, circular arc, trapezoid, and permutation graphs.

Tight logarithmic lower bounds on certificate sizes for these classes.

Proof labeling schemes effectively recognize these classes in distributed settings.

Abstract

In the context of distributed certification, the recognition of graph classes has started to be intensively studied. For instance, different results related to the recognition of planar, bounded tree-width and $H$-minor free graphs have been recently obtained. The goal of the present work is to design compact certificates for the local recognition of relevant geometric intersection graph classes, namely interval, chordal, circular arc, trapezoid and permutation. More precisely, we give proof labeling schemes recognizing each of these classes with logarithmic-sized certificates. We also provide tight logarithmic lower bounds on the size of the certificates on the proof labeling schemes for the recognition of any of the aforementioned geometric intersection graph classes.