New Distributed Interactive Proofs for Planarity: A Matter of Left and Right

We provide new distributed interactive proofs (DIP) for planarity and related graph families. The notion of a \emph{distributed interactive proof} (DIP) was introduced by Kol, Oshman, and Saxena (PODC 2018). In this setting, the verifier consists of $n$ nodes connected by a communication graph $G$. The prover is a single entity that communicates with all nodes by short messages. The goal is to verify that the graph $G$ satisfies a certain property (e.g., planarity) in a small number of rounds, and with a small communication bound, denoted as the \emph{proof size}. Prior work by Naor, Parter and Yogev (SODA 2020) presented a DIP for planarity that uses three interaction rounds and a proof size of $O(\log n)$. Feuilloley et al.\ (PODC 2020) showed that the same can be achieved with a single interaction round and without randomization, by providing a proof labeling scheme with a proof size of $O(\log n)$. In a subsequent work, Bousquet, Feuilloley, and Pierron (OPODIS 2021) achieved the same bound for related graph families such as outerplanarity, series-parallel graphs, and graphs of treewidth at most $2$. In this work, we design new DIPs that use exponentially shorter proofs compared to the state-of-the-art bounds.