3-连通1-平面图的正交多边形可见性表示
Ortho-polygon Visibility Representations of 3-connected 1-plane Graphs
摘要：
$1$平面图$G$的正交多边形可见性表示$Γ$（$G$的OPVR）是嵌入保持绘图，其将$G$的每个顶点映射到不同的正交多边形，并将$G$的每个边映射到其端顶点之间的垂直或水平可见性。如果$Γ$的每个多边形最多有$k$反射角，则表示$Γ$具有顶点复杂度$k$。已知$3$ -连通$1$ -平面图允许顶点复杂度至多为12的OPVR，而在某些情况下可能需要顶点复杂度至少为2.在本文中，我们通过证明顶点复杂度为5总是足够的，而在某些情况下可能需要顶点复杂度为4来缩小这一差距。这些结果是基于对$3$ -连通$1$ -平面图中B-、T-和W-构型的组合性质的研究。上界的一个含义是存在一个$\tilde{O}(n^\frac{10}{7})$ -时间绘图算法，该算法计算大小为$O(n) \times O(n)$且顶点复杂度至多为5的整数网格上的$n$ -顶点$3$ -连通$1$ -平面图的OPVR.
英文摘要：
An ortho-polygon visibility representation $Γ$ 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 $Γ$ has vertex complexity $k$ if every polygon of $Γ$ 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.
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