A Planarity Criterion for Graphs
Kosta Došen, Zoran Petrić · SIAM Journal on Discrete Mathematics · 2015
It is proved that a connected graph is planar if and only if all its cocycles with at least four edges are “grounded” in the graph. The notion of grounding of this planarity criterion, which is purely combinatorial, stems from the intuitive idea that with planarity, there should be a linear ordering of the edges of a cocycle such that in the two subgraphs remaining after the removal of these edges, there can be no crossing of disjoint paths that join the vertices of these edges. The proof given in the paper of the right-to-left direction of the equivalence is based on Kuratowski's theorem for planarity involving $K_{3,3}$ and $K_5$, but the criterion itself does not mention $K_{3,3}$ and $K_5$. Some other variants of the criterion are also shown necessary and sufficient for planarity.