  Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence
of a \emph{distributed interactive proof} for planarity (i.e., for certifying
that a network is planar), using a sophisticated generic technique for
constructing distributed IP protocols based on sequential IP protocols. The
interactive proof for planarity is based on a distributed certification of the
correct execution of any given sequential linear-time algorithm for planarity
testing. It involves three interactions between the prover and the randomized
distributed verifier (i.e., it is a \dMAM\/ protocol), and uses small
certificates, on $O(\log n)$ bits in $n$-node networks. We show that a single
interaction from the prover suffices, and randomization is unecessary, by
providing an explicit description of a \emph{proof-labeling scheme} for
planarity, still using certificates on just $O(\log n)$ bits. We also show that
there are no proof-labeling schemes -- in fact, even no \emph{locally checkable
proofs} -- for planarity using certificates on $o(\log n)$ bits.
