Let $\{C(G)\}$ be the set of chordless cycles of a graph $G$. Compare the cycles pairwise. Let $\{V\}$ represent the pairs which have exactly one vertex in common; and, let $\{P\}$ represent those pairs which have a single continuous sequence of edges, ie, a path, in common. For a pair in $\{V\}$, let $X$ denote their common vertex. Obviously, for planarity, the edges at vertex $X$ must be ordered so that the cycles do not overlap, ie., crossover. Similar restrictions obtain on the orderings at the two terminal vertices for the common path shared by a pair of cycles in $\{P\}$.

In either type of conjunction, certain edge orderings on common vertices are required for planarity of $G$. Have the Kuratowski criteria for planarity been shown to be equivalent to the two types of restrictions described above?