The 3-Steiner root problem

  • Authors:

  • Maw-Shang Chang;Ming-Tat Ko

  • Affiliations:

  • Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi, Taiwan, R.O.C.;Institute of Information Science, Academia Sinica, Taipei, Taiwan, R.O.C.

  • Venue:
  • WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
  • Year:
  • 2007

Quantified Score

Hi-index 0.00

Visualization

Abstract

For a graph G and a positive integer k, the k-power of G is the graph Gk with V(G) as its vertex set and {(u,v)|u, v ∈ V(G), dG (u, v) ≤ k} as its edge set where dG(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 Tk 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 TLk, with TLk = (V,E), where E = {(u, v) | u, v ∈ V and dT(u, v) ≤ k}. We call T a k-leaf root of TLk. 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.