Minimum survivable graphs with bounded distance increase

We study in graphs a property related to fault-tolerance in case a node fails. A graph G is k-self-repairing, where k is a non-negative integer, if after the removal of any vertex no distance in the surviving graph increase by more than k. We give upper and lower bounds on the minimum number of edges of a k-self-repairing graph for prescribed k and n, where n is the order of the graph. We also prove that the problem of finding, in a k-self-repairing graph, a spanning k-self-repairing subgraph of the minimum size is NP-Hard.