We provide new communication-efficient distributed interactive proofs for planarity. The notion of a \emph{distributed interactive proof (DIP)} was introduced by Kol, Oshman, and Saxena (PODC 2018). In a DIP, the \emph{prover} is a single centralized entity whose goal is to prove a certain claim regarding an input graph $G$. To do so, the prover communicates with a distributed \emph{verifier} that operates concurrently on all $n$ nodes of $G$. A DIP is measured by the amount of prover-verifier communication it requires. Namely, the goal is to design a DIP with a small number of interaction rounds and a small \emph{proof size}, i.e., a small amount of communication per round. Our main result is an $O(\log ^{*}n)$-round DIP protocol for embedded planarity and planarity with a proof size of $O(1)$ and $O(\lceil\log \Delta/\log ^{*}n\rceil)$, respectively. In fact, this result can be generalized as follows. For any $1\leq r\leq \log^{*}n$, there exists an $O(r)$-round protocol for embedded planarity and planarity with a proof size of $O(\log ^{(r)}n)$ and $O(\log ^{(r)}n+\log \Delta /r)$, respectively.

翻译：我们为平面性判定问题提出了新型通信高效的分布式交互证明。\emph{分布式交互证明（DIP）}的概念由Kol、Oshman和Saxena（PODC 2018）首次提出。在DIP框架中，\emph{证明者}是集中式的单一实体，其目标是证明关于输入图$G$的特定性质。为此，证明者需与分布式\emph{验证者}进行通信，该验证者同时在$G$的所有$n$个节点上运行。DIP的性能通过证明者-验证者间所需的通信量来衡量，具体目标是设计具有较少交互轮次和较小\emph{证明规模}（即每轮通信量）的DIP协议。我们的核心成果是：针对嵌入平面性和平面性判定问题，分别构建了证明规模为$O(1)$和$O(\lceil\log \Delta/\log ^{*}n\rceil)$的$O(\log ^{*}n)$轮DIP协议。该结果可进一步推广为：对于任意$1\leq r\leq \log^{*}n$，存在针对嵌入平面性和平面性判定的$O(r)$轮协议，其证明规模分别为$O(\log ^{(r)}n)$和$O(\log ^{(r)}n+\log \Delta /r)$。