Before revision:  The results are based on the study of the combinatorial properties of the B-, T-, and W-configurations in a 3-connected 1-plane graph . An implication of the upper bound is the existence of an O(n^10{7})-time drawing algorithm that computes an OPVR of an n-vertex 3-connected 1-plane graph on an integer grid of size O(n) \times O(n) and with vertex complexity at most five.
After revision:  These results are based on the study of the combinatorial properties of the B-, T-, and W-configurations in a 3-connected 1-plane graph . An implication of the upper bound is the existence of an O(n^10{7})-time drawing algorithm that computes an OPVR of an n-vertex 3-connected 1-plane graph on an integer grid of size O(n) \times O(n) and with vertex complexity at most five.