Compact distributed certification of geometric graph classes

Distributed proofs allow network nodes to collectively verify if the network satisfies a given predicate. The most versatile mechanism, known as a proof labeling scheme (PLS), functions as the distributed equivalent of NP, where a non-trustable prover assigns each node a certificate. Nodes exchange these certificates with their neighbors to prove the graph satisfies the predicate, with the certificate size being the primary complexity measure. Many graph properties, like planarity or bounded tree-width, can be certified with O(log⁡n)-bit certificates on n-node graphs. This paper presents O(log⁡n) distributed certifications for recognizing geometric graph classes commonly found in distributed systems: interval graphs, chordal graphs, circular arc graphs, trapezoid graphs, and permutation graphs. It also establishes tight lower bounds on the certificate sizes required for these geometric intersection graph classes, proving that the proposed certifications are optimal.