具有自同构的容错超图

摘要：

给定$n$顶点上的图$Y$和期望的容错水平$k$，容错系统设计中的目标是在$n + k$顶点上构造超图$X$，使得从$X$移除任何$k$节点留下包含$Y$的图。为了在故障发生时围绕故障重新配置，还要求$X$的$k$节点的任何两个子集处于其自同构群的作用的相同轨道中。本文证明了这样的超图一定是完全图。这意味着拥有一个具有$k$容错性并且还支持自守重新配置的互连网络非常昂贵。我们的工作解决了文献中的一个公开问题。证明使用的结果由于卡梅隆对$k$ -齐次群。

英文摘要：

Given a graph $Y$ on $n$ vertices and a desired level of fault-tolerance $k$, an objective in fault-tolerant system design is to construct a supergraph $X$ on $n + k$ vertices such that the removal of any $k$ nodes from $X$ leaves a graph containing $Y$. In order to reconfigure around faults when they occur, it is also required that any two subsets of $k$ nodes of $X$ are in the same orbit of the action of its automorphism group. In this paper, we prove that such a supergraph must be the complete graph. This implies that it is very expensive to have an interconnection network which is $k$-fault-tolerant and which also supports automorphic reconfiguration. Our work resolves an open problem in the literature. The proof uses a result due to Cameron on $k$-homogeneous groups.

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。