论文标题
哪些图在$ \ ell_p^d $中是刚性的？
Which graphs are rigid in $\ell_p^d$?
论文作者
论文摘要
我们提出了三个结果，这些结果支持以下猜想：图在$ d $ d $ d $ \ ell_p $ -space中是最小的，其中$ p \ in（1，\ infty）$和$ p \ not = 2 $，仅当它是$（d，d，d，d）$ - tigh。首先，我们介绍了一个图形支撑操作，该操作从$ \ ell_p^d $传递到$ \ ell_p^{d+1} $时，该操作可保留一般刚性的独立性。然后，我们证明，每$（d，d）$ - 最低学位的稀疏图最多$ d+1 $和最高学位最多$ d+2 $在$ \ ell_p^d $中是独立的。最后，我们证明，投影平面的每个三角剖分都在$ \ ell_p^3 $中最少。还为更一般的严格凸和平滑规范空间提供了刚性的刚度保留图表的目录，我们证明球体的每个三角剖分都是独立于该类别的3维空间。
We present three results which support the conjecture that a graph is minimally rigid in $d$-dimensional $\ell_p$-space, where $p\in (1,\infty)$ and $p\not=2$, if and only if it is $(d,d)$-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from $\ell_p^d$ to $\ell_p^{d+1}$. We then prove that every $(d,d)$-sparse graph with minimum degree at most $d+1$ and maximum degree at most $d+2$ is independent in $\ell_p^d$. Finally, we prove that every triangulation of the projective plane is minimally rigid in $\ell_p^3$. A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.
