Planar Graphs: Logical Complexity and Parallel Isomorphism Tests
<|reference_start|>Planar Graphs: Logical Complexity and Parallel Isomorphism Tests: We prove that every triconnected planar graph is definable by a first order sentence that uses at most 15 variables and has quantifier depth at most $11\log_2 n+43$. As a consequence, a canonic form of such graphs is computable in $AC^1$ by the 14-dimensional Weisfeiler-Lehman algorithm. This provides another way to show that the planar graph isomorphism is solvable in $AC^1$.<|reference_end|>
@article{verbitsky2006planar,
    title={Planar Graphs: Logical Complexity and Parallel Isomorphism Tests},
    author={Oleg Verbitsky},
    journal={arXiv preprint arXiv:cs/0607033},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0607033},
    primaryClass={cs.CC cs.LO}
}