满足奇偶约束的平面图的平面扩充
Plane augmentation of plane graphs to meet parity constraints
摘要：
平面拓扑图$G=(V,E)$是在平面中绘制的图，其顶点是平面中的点，并且其边是除了端点之外不相交的简单曲线。给定一个平面拓扑图$G=(V,E)$和一个奇偶约束集$C_G$，其中每个顶点都在其度上分配了奇偶约束，无论是偶数还是奇数，如果在同一个顶点集上存在一个平面拓扑图$H$，使得$G$和$H$边不相交，并且它们的并集是一个满足所有奇偶约束的平面拓扑图，则称$G$是\emph{topologically augmentable}满足$C_G$.本文证明了判定平面拓扑图是否可拓扑扩张以满足奇偶约束的问题是$\mathcal{NP}$ -完全的，即使必须改变其奇偶性的顶点集是$V$或奇数度的顶点集.特别地，判定平面拓扑图是否可以扩充为欧拉平面拓扑图是$\mathcal{NP}$ -完全的。类似的复杂性的结果，得到的扩增时，必须做一个平面拓扑完美匹配的顶点之间不满足他们的奇偶校验。我们将这些硬度结果推广到平面图，当增广图必须是平面的，和平面几何图（平面拓扑图，其边缘是直线段）。此外，当要求扩充是由不满足奇偶性的顶点之间的平面几何完美匹配完成时，我们还证明了对于平面几何树和路径，这个扩充问题是$\mathcal{NP}$ -完全的.对于特定的家庭的最大外平面图，我们刻画了最大外平面图的拓扑扩充，以满足一组奇偶约束。
英文摘要：
A plane topological graph $G=(V,E)$ is a graph drawn in the plane whose vertices are points in the plane and whose edges are simple curves that do not intersect, except at their endpoints. Given a plane topological graph $G=(V,E)$ and a set $C_G$ of parity constraints, in which every vertex has assigned a parity constraint on its degree, either even or odd, we say that $G$ is \emph{topologically augmentable} to meet $C_G$ if there exits a plane topological graph $H$ on the same set of vertices, such that $G$ and $H$ are edge-disjoint and their union is a plane topological graph that meets all parity constraints. In this paper, we prove that the problem of deciding if a plane topological graph is topologically augmentable to meet parity constraints is $\mathcal{NP}$-complete, even if the set of vertices that must change their parities is $V$ or the set of vertices with odd degree. In particular, deciding if a plane topological graph can be augmented to a Eulerian plane topological graph is $\mathcal{NP}$-complete. Analogous complexity results are obtained, when the augmentation must be done by a plane topological perfect matching between the vertices not meeting their parities. We extend these hardness results to planar graphs, when the augmented graph must be planar, and to plane geometric graphs (plane topological graphs whose edges are straight-line segments). In addition, when it is required that the augmentation is made by a plane geometric perfect matching between the vertices not meeting their parities, we also prove that this augmentation problem is $\mathcal{NP}$-complete for plane geometric trees and paths. For the particular family of maximal outerplane graphs, we characterize maximal outerplane graphs that are topological augmentable to satisfy a set of parity constraints.
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