An ortho-polygon visibility representation \Gamma of a 1-plane graph G (OPVR of G) is an embedding preserving drawing that maps each vertex of G to a distinct orthogonal polygon and each edge of G to a vertical or horizontal visibility between its end-vertices. The representation \Gamma has vertex complexity k if every polygon of \Gamma has at most k reflex corners. It is known that if G is 3-connected it admits an OPVR with vertex complexity at most twelve, while a vertex complexity at least two may be required in some cases. In this paper  we reduce this gap by showing that vertex complexity five is always sufficient, while vertex complexity four may be required in some cases. 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.
An ortho-polygon visibility representation \Gamma of a 1-plane graph G (OPVR of G) is an embedding preserving drawing that maps each vertex of G to a distinct orthogonal polygon and each edge of G to a vertical or horizontal visibility between its end-vertices. The representation \Gamma has vertex complexity k if every polygon of \Gamma has at most k reflex corners. It is known that 3-connected 1-plane graphs admit an OPVR with vertex complexity at most twelve, while vertex complexity at least two may be required in some cases. In this paper , we reduce this gap by showing that vertex complexity five is always sufficient, while vertex complexity four may be required in some cases. These results are based on the study of the combinatorial properties of the B-, T-, and W-configurations in 3-connected 1-plane graphs . An implication of the upper bound is the existence of a 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.
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