哪些图在 $\ell_p^d$ 中是刚性的？
Which graphs are rigid in $\ell_p^d$?
AI总结：
本文通过支撑操作、度受限稀疏图独立性及射影平面三角剖分最小刚性等结果，支持图在 $\ell_p^d$ 中最小刚性等价于 $(d,d)$-紧的猜想。
摘要：
我们给出三项结果，支持如下猜想：当 $p\in(1,\infty)$ 且 $p\not=2$ 时，一个图在 $d$ 维 $\ell_p$ 空间中是最小刚性的，当且仅当它是 $(d,d)$-紧的。首先，我们引入一种图支撑操作，该操作在从 $\ell_p^d$ 过渡到 $\ell_p^{d+1}$ 时保持一般刚性拟阵中的独立性。随后，我们证明每个最小度至多为 $d+1$、最大度至多为 $d+2$ 的 $(d,d)$-稀疏图在 $\ell_p^d$ 中都是独立的。最后，我们证明射影平面的每个三角剖分在 $\ell_p^3$ 中都是最小刚性的。对于更一般的严格凸且光滑的赋范空间类，我们还提供了一个保持刚性的图移动目录，并证明球面的每个三角剖分在该类中的三维空间里都是独立的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
