The 3-Steiner Root Problem

Abstract

For a graph G and a positive integer k, the k-power of G is the graph G ^k with V(G) as its vertex set and {(u,v)| u, v ∈ V(G),  d _G (u, v) ≤ k} as its edge set where d _G (u,v) is the distance between u and v in graph G. The k-Steiner root problem on a graph G asks for a tree T with V(G) ⊆ V(T) and G is the subgraph of T ^k induced by V(G). If such a tree T exists, we call it a k-Steiner root of G. This paper gives a linear time algorithm for the 3-Steiner root problem. Consider an unrooted tree T with leaves one-to-one labeled by the elements of a set V. The k-leaf power of T is a graph, denoted $T_{L}^{k}$, with $T_{L}^{k} = (V, E)$, where E = {(u,v) |u, v ∈ V   and   d _T (u,v) ≤ k}. We call T a k-leaf root of $T_{L}^{k}$. The k-leaf power recognition problem is to decide whether a graph has such a k-leaf root. The complexity of this problem is still open for k ≥ 5 [6]. It can be solved in polynomial time if the (k − 2)-Steiner root problem can be solved in polynomial time [6]. Our result implies that the k-leaf power recognition problem can be solved in linear time for k = 5.