图的退化单位距离表示的计算复杂性
On the computational complexity of degenerate unit distance representations of graphs
摘要：
有些图允许以这样一种（自然的）方式在欧几里德k空间中绘图，即边被表示为单位长度的线段。这样的绘图将被称为k维单位距离表示。当两个非邻顶点画在同一点上时，我们说这个表示是退化的。图的维数（欧几里德维数）被定义为给定图具有非退化k维单位距离表示（具有非邻顶点映射到距离不为1的点的性质）所需的最小整数k。证明了判定一个输入图是否同态于一个维数k ≥ 2的图（其中欧几里德维数k ≥ 2）是NP-难问题。
英文摘要：
Some graphs admit drawings in the Euclidean k-space in such a (natu- ral) way, that edges are represented as line segments of unit length. Such drawings will be called k dimensional unit distance representations. When two non-adjacent vertices are drawn in the same point, we say that the representation is degenerate. The dimension (the Euclidean dimension) of a graph is defined to be the minimum integer k needed that a given graph has non-degenerate k dimensional unit distance representation (with the property that non-adjacent vertices are mapped to points, that are not distance one appart). It is proved that deciding if an input graph is homomorphic to a graph with dimension k >= 2 (with the Euclidean dimension k >= 2) are NP-hard problems.
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