k-Neighborhood-Covering and -Independence Problems for Chordal Graphs

Suppose G = (V,E) is a simple graph and k is a fixed positive integer. A vertex z k-neighborhood-covers an edge (x, y) if d(z, x) ≤ k and d(z, y) ≤ k. A k-neighborhoodcovering set is a set C of vertices such that every edge in E is k-neighborhood-covered by some vertex in C. A k-neighborhood-independent set is a set of edges in which no two distinct edges can be k-neighborhood-covered by the same vertex in V . In this paper we first prove that the kneighborhood-covering and the k-neighborhood-independence problems are NP-complete for chordal graphs. We then present a linear-time algorithm for finding a minimum k-neighborhood-covering set and a maximum k-neighborhood-independent set for a strongly chordal graph provided that a strong elimination ordering is given in advance.