Planarity can be Verified by an Approximate Proof Labeling Scheme in Constant-Time

TL;DR

This paper demonstrates that planarity and similar properties of bounded-degree graphs can be verified efficiently using approximate proof labeling schemes in constant time, regardless of graph size.

Contribution

It introduces the first constant-time approximate proof labeling scheme for verifying planarity and related graph properties.

Findings

Planarity can be verified in constant time using approximate proof labeling schemes.

The scheme applies to bounded-degree planar, outer-planar, and bounded genus graphs.

Verification does not depend on the size of the graph.

Abstract

Approximate proof labeling schemes were introduced by \\Censor-Hillel, Paz and Perry \cite{CPP}. Roughly speaking, a graph property~$\cP$ can be verified by an approximate proof labeling scheme in constant-time if the vertices of a graph having the property can be convinced, in a short period of time not depending on the size of the graph, that they are having the property $\cP$ or at least they are not far from being having the property $\cP$. The main result of this paper is that bounded-degree planar graphs (and also outer-planar graphs, bounded genus graphs, knotlessly embeddable graphs etc.) can be verified by an approximate proof labeling scheme in constant-time.