chordless cycles and planarity in graphs

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},

In either type of conjunction, certain edge orderings on common vertices are required for planarity of G. Can the Kuratowski criteria for planarity be so determined?