Ortho-polygon visibility representations of embedded graphs

An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. Namely, we prove that if G is 1-plane (i.e., it has at most one crossing per edge) an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute in O(n) time an OPVR of G whose vertex complexity is bounded by a constant. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be Ω(n). In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed. Finally, we present the results of an experimental study on the vertex complexity of OPVRs of 1-plane graphs.