Compact Distributed Certification of Planar Graphs - Feuilloley, Laurent; Fraigniaud, Pierre; Montealegre, Pedro; Rapaport, Ivan; Rémila, Éric; Todinca, Ioan (Algorithmica, 2021)
Naor M., Parter M., Yogev E.: (The power of distributed verifiers in interactive proofs. In: 31st ACM-SIAM symposium on discrete algorithms (SODA), pp 1096–115, 2020. https://doi.org/10.1137/1.9781611975994.67 ) have recently demonstrated the existence of a 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 with the prover suffices, and randomization is unecessary, by providing an explicit description of a 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 locally checkable proofs —for planarity using certificates on o ( log n ) bits.