平面直线图的最小权连通性扩充
Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs
摘要：
我们考虑边插入和删除操作，增加给定的平面直线图（PSLG）的连通性，同时最小化输出的总边长。我们证明了在一般位置上的每个连通PSLG $G=(V,E)$都可以通过增加新的总欧氏长度为$\|E^+\|\leq 2\|E\|$的边而扩充为2-连通PSLG $(V,E\cup E^+)$，并且这个界是最好的可能界.最优边集$E^+$可以在$O(|V|^4)$时间内计算;然而，当$G$断开时，问题变为NP难的。此外，存在将连接的PSLG $G=(V,E)$转换成平面直线循环$G'=(V,E')$的边插入和删除序列，使得$\|E'\|\leq 2\|{\rm MST}(V)\|$，并且图在所有阶段保持连接，其中边长度低于$\|E\|+\|{\rm MST}(V)\|$。这些边界是最好的可能。
英文摘要：
We consider edge insertion and deletion operations that increase the connectivity of a given planar straight-line graph (PSLG), while minimizing the total edge length of the output. We show that every connected PSLG $G=(V,E)$ in general position can be augmented to a 2-connected PSLG $(V,E\cup E^+)$ by adding new edges of total Euclidean length $\|E^+\|\leq 2\|E\|$, and this bound is the best possible. An optimal edge set $E^+$ can be computed in $O(|V|^4)$ time; however the problem becomes NP-hard when $G$ is disconnected. Further, there is a sequence of edge insertions and deletions that transforms a connected PSLG $G=(V,E)$ into a planar straight-line cycle $G'=(V,E')$ such that $\|E'\|\leq 2\|{\rm MST}(V)\|$, and the graph remains connected with edge length below $\|E\|+\|{\rm MST}(V)\|$ at all stages. These bounds are the best possible.
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