Ortho-polygon Visibility Representations of 3-connected 1-plane Graphs

TL;DR

This paper improves the known bounds on the complexity of ortho-polygon visibility representations for 3-connected 1-plane graphs, showing that vertex complexity five always suffices and providing an efficient drawing algorithm.

Contribution

It establishes that vertex complexity five is always sufficient for OPVRs of 3-connected 1-plane graphs, reducing the previous upper bound from twelve.

Findings

Vertex complexity five always suffices for 3-connected 1-plane graphs.

An algorithm computes OPVRs with vertex complexity at most five in near-linear time.

The algorithm produces drawings on an integer grid of size O(n) x O(n).

Abstract

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 $\tilde{O}(n^\frac{10}{7})$-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.