Local certification of geometric graph classes

TL;DR

These bounds are tight up to a constant factor and give the first known examples of hereditary (and even monotone) graph classes for which the certificates must have linear size.

Abstract

The goal of local certification is to locally convince the vertices of a graph $G$ that $G$ satisfies a given property. A prover assigns short certificates to the vertices of the graph, then the vertices are allowed to check their certificates and the certificates of their neighbors, and based only on this local view, they must decide whether $G$ satisfies the given property. If the graph indeed satisfies the property, all vertices must accept the instance, and otherwise at least one vertex must reject the instance (for any possible assignment of certificates). The goal is to minimize the size of the certificates. In this paper we study the local certification of geometric and topological graph classes. While it is known that in $n$-vertex graphs, planarity can be certified locally with certificates of size $O(\log n)$, we show that several closely related graph classes require certificates of size $Ω(n)$. This includes penny graphs, unit-distance graphs, (induced) subgraphs of the square grid, 1-planar graphs, and unit-square graphs. These bounds are tight up to a constant factor and give the first known examples of hereditary (and even monotone) graph classes for which the certificates must have linear size. For unit-disk graphs we obtain a lower bound of $Ω(n^{1-δ})$ for any $δ>0$ on the size of the certificates, and an upper bound of $O(n \log n)$. The lower bounds are obtained by proving rigidity properties of the considered graphs, which might be of independent interest.

Figures (16)

• Figure 1: Triangle-free intersection graphs of unit-disks and unit-squares in the plane, and the associated planar graph embeddings.
• Figure 2: Construction of $L,R$ and $g$ for penny graphs in the case where $N=2$, with $A,B\subseteq\{1, \ldots, N\}$. Color red highlights vertices and edges that depend on the choice of $A$, and color blue highlights vertices and edges that depend on the choice of $B$.
• Figure 3: Construction of $L,R$ and $g$ for subgraphs of the square grid in the case where $N=3$, with $A,B\subseteq\{1, \ldots, N\}$. Color red highlights vertices and edges that depend on the choice of $A$, and color blue highlights vertices and edges that depend on the choice of $B$.
• Figure 4: The construction of $L$, $R$ and $g$ for 1-planar graphs in the case where $N=5$, with $A,B\subseteq \{1, \ldots, N\}$. Color red highlights edges that depend on the choice of $A$, and color blue highlights edges that depend on the choice of $B$.
• Figure 5: A vertex of degree 4 in a triangle-free unit-square graph.

Theorems & Definitions (60)

• Proposition 2.1
• proof
• Theorem 2.2: Whi32
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Corollary 3.3
• Theorem 4.1
• proof