论文标题
图形上的地球几何形状
Geodesic Geometry on Graphs
论文作者
论文摘要
我们研究了地球几何形状的图理论类似物。在图$ g =（v，e）$中，我们考虑了一个路径系统$ \ mathcal {p} = \ {p_ {p_ {u，v} | u，v \ in v \} $，其中$ p_ {u，v} $连接Vertices $ u $ u $和$ v $。该系统是一致的，因为如果顶点$ y，z $在$ p_ {u，v} $中，则它们之间的子路径与$ p_ {u，v} $之间的子路径与$ p_ {y，z} $一致。地图$ w：e \ to（0，\ infty）$被认为会诱导$ \ mathcal {p} $，如果每个$ u，v \ in v $ in v $ p $ p_ {u，v} $是$ w $ geodesic。我们说，如果每种一致的路径系统都是由一些这样的$ W $引起的，则$ G $是可以METRIA的。正如我们所表明的那样，可计算图非常罕见，而存在无限的许多$ 2 $连接的可元图。
We investigate a graph theoretic analog of geodesic geometry. In a graph $G=(V,E)$ we consider a system of paths $\mathcal{P}=\{P_{u,v}|u,v\in V\}$ where $P_{u,v}$ connects vertices $u$ and $v$. This system is consistent in that if vertices $y, z$ are in $P_{u,v}$, then the sub-path of $P_{u,v}$ between them coincides with $P_{y,z}$. A map $w: E\to(0,\infty)$ is said to induce $\mathcal{P}$ if for every $u, v\in V$ the path $P_{u,v}$ is $w$-geodesic. We say that $G$ is metrizable if every consistent path system is induced by some such $w$. As we show, metrizable graphs are very rare, whereas there exist infinitely many $2$-connected metrizable graphs.
