Local certification of graph decompositions and applications to minor-free classes

TL;DR

This paper develops new tools for local certification of graph classes and proves that minor-free graphs can be certified with logarithmic-sized labels, extending previous results to a broader class of graphs.

Contribution

The paper introduces new decomposition methods for graph certification and demonstrates that minor-free graphs can be certified with small labels, answering an open question.

Findings

Minor-free graphs can be certified with O(log n) labels.

Matching lower bounds establish the optimality of the certification size.

New decomposition tools facilitate certification of complex graph classes.

Abstract

Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph $G$ belongs to a given graph class~$\mathcal{G}$. Such certifications with labels of size $O(\log n)$ (where $n$ is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors. In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor $H$, $H$-minor-free graphs can indeed be certified with labels of size $O(\log n)$. We also show matching lower bounds with a simple new proof technique.