A graph G* is a k-node fault-tolerant supergraph of a graph G, denoted k-NFT (G), if every graph obtained by removing k nodes from G* contains G. A k-NFT (G) graph G* is said to be optimal if it contains n + k nodes, where n is the number of nodes of G and G* has the minimum number of edges among all (n + k)-node k-NFT supergraphs of G. We survey prior results on the design of optimal k-NFT supergraphs of various useful forms of G; this work covers cycles and various types of trees. We also introduce the concept of exact node fault tolerance, which requires that every graph obtained by removing k nodes from G* be isomorphic to G, and explore its basic properties. We conclude with a discussion of some unsolved and partially solved problems.
