Largest Component and Node Fault Tolerance for Grids
A graph $G$ is called $t$-node fault tolerant with respect to $H$ if $G$ still contains a subgraph isomorphic to $H$ after removing any $t$ of its vertices. The least value of $|E(G)|-|E(H)|$ among all such graphs $G$ is denoted by $\Delta(t,H)$. We study fault tolerance with respect to some natural architectures of a computer network, i.e. the $d$-dimensional toroidal grids and the hypercubes. We provide the first non-trivial lower bounds for $\Delta(1,H)$ in these cases. For this aim we establish a general connection between the notion of fault tolerance and the size of a largest component of a graph. In particular, we give for all values of $k$ (and $n$) a lower bound on the order of the largest component of any graph obtained from $C_n\Box C_n$ via removal of $k$ of its vertices, which is in general optimal.