We know that in graph theory, an independent set is a set of vertices, such that no two of which are adjacent. There is rich theory about independent set, including approximation algorithm for finding maximal independent set, estimation of the size of the maximal independent set in random graphs, etc.
I wonder whether we can extend the definition, and define second-order independent set - any two vertices of which have distance at least 3? I.e., they can not have shared neighbor. Moreover, can we derive corresponding theory for such second-order independent set?