Property Testing of Planarity in the CONGEST model

We give a distributed algorithm in the {\sf CONGEST} model for property testing of planarity with one-sided error in general (unbounded-degree) graphs. Following Censor-Hillel et al. (DISC 2016), who recently initiated the study of property testing in the distributed setting, our algorithm gives the following guarantee: For a graph $G = (V,E)$ and a distance parameter $\epsilon$, if $G$ is planar, then every node outputs {\sf accept\/}, and if $G$ is $\epsilon$-far from being planar (i.e., more than $\epsilon\cdot |E|$ edges need to be removed in order to make $G$ planar), then with probability $1-1/{\rm poly}(n)$ at least one node outputs {\sf reject}. The algorithm runs in $O(\log|V|\cdot{\rm poly}(1/\epsilon))$ rounds, and we show that this result is tight in terms of the dependence on $|V|$. Our algorithm combines several techniques of graph partitioning and local verification of planar embeddings. Furthermore, we show how a main subroutine in our algorithm can be applied to derive additional results for property testing of cycle-freeness and bipartiteness, as well as the construction of spanners, in minor-free (unweighted) graphs.