Compact Distributed Certification of Planar Graphs

Abstract

Naor, Parter, and Yogev (SODA 2020) 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 from 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.

Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs

“…Also, the structural properties of distance-hereditary graphs have been used in the design of compact routing tables for interconnection networks [5]. Regarding local certification, other results on the recognition of graph classes include planar graphs [14] and graphs with bounded genus [13], where the authors showed that both classes admit proof-labeling schemes with logarithmic-sized certificates. Recently, Naor, Parter and Yogev defined in [35] a compiler which (1) turns any problem solved in NP in time τ (n) into a dMAM protocol using private randomness and bandwidth τ (n) log n/n and; (2) turns any problem which can be solved in NC into a dAM protocol with private randomness, poly log n rounds of interaction and bandwidth poly log n. For example, this result implies that, any class of sparse graphs that can be recognized in linear time, can also be recognized by a dMAM protocol with logarithmic-sized certificates.…”