A Generalization of maximal independent sets Original Research Article

We generalize the concept of maximal-independent set in the following way. For a nonnegative integer k we define a k-insulated set of a graph G as a subset S of its vertices such that each vertex in S is adjacent to at most k other vertices in S and each vertex not in S is adjacent to at least k+1 vertices in S. We show that it is NP-hard to approximate a maximum k-insulated set within a polynomial factor and describe a polynomial algorithm which approximates a maximum k-insulated set in an n-vertex graph to within the factor of cnk/log2n, for a constant c>0. We also give an O(kn2) algorithm which finds an arbitrary k-insulated set.