The Maximum k-Dependent and f-Dependent Set Problem

Let k be a positive integer. A k-dependent set in an undirected graph G = (V,E) is a subset of the set V of vertices such that no vertex in the subset is adjacent to more than k vertices of the subset. This subset induces a subgraph of G of maximum degree bounded by k. A 0-dependent set in G is simply an independent set of vertices in G. Furthermore, an 1-dependent set is in general a set of independent vertices and edges and a 2-dependent set is a set of independent paths and cycles. The problem of constructing a maximum k-dependent set and its decision version have been studied in [6]. The NP-completeness of the decision version has been shown for arbitrary graphs and each k _> 0. On the other hand a linear-time algorithm for the construction problem restricted to trees has been presented in [6]. Furthermore, for each constant k the problem of finding a maximum k-dependent set for a graph with constant treewidth has been observed to be solvable in linear time. The problem of finding a maximmn k-dependent set for k = 2 has several applications, for example in information dissemination in hypercubes with a large number of faulty processors [:~]. A generalization of this problem called the maximum f-dependeut set problem has been given in [7]. Given weights f(v) E ]No for v E V, an f-dependent set is a subset A of V such that each vertex v E A is adjacent to at most f(v) vertices in A. In [7] parallel algorithms for finding maximal k and f-dependent sets have been given. In this paper we analyze both problems for bipartite graphs, cographs, trees, split graphs and graphs with bounded treewidth. Among others, we show that the decision version of the maximum k-dependent set problem restricted to planar, bipartite graphs is NP-complete for any given k >_ 1. This contrast with the well known fact that the maxinmm 0-dependent (i.e. independent) set in a bipartite graph can be found in polynomial time by reduction to maximum matching (see [10]) via KSnig-Egervary theorem (see [15] and [8]). Next, we give polynomial algorithms for both problems restricted to cographs, trees and graphs with bounded treewidth. On the other hand, we show that the complexity differs for split graphs; we give a polynomial time algorithm for the maximum k-dependent set problem and show the NP-completeness for