Bounded degree closest k-tree power is NP-complete

Abstract

An undirected graph G = (V, E) is the k-power of an undirected tree T = (V, E′) if (u, v) ∈ E iff u and v are connected by a path of length at most k in T. The tree T is called the tree root of G. Tree powers can be recognized in polynomial time. The thus naturally arising question is whether a graph G can be modified by adding or deleting a specified number of edges such that G becomes a tree power. This problem becomes NP-complete for k ≥ 2. Strengthening this result, we answer the main open question of Tsukiji and Chen [COCOON 2004] by showing that the problem remains NP-complete when additionally demanding that the tree roots must have bounded degree. © Springer-Verlag Berlin Heidelberg 2005.