On tree 3-spanners in directed path graphs

  • Authors:

  • B. S. Panda;Anita Das

  • Affiliations:

  • Computer Science and Application Group, Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India;Computer Science and Application Group, Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

  • Venue:
  • Networks
  • Year:
  • 2007

Quantified Score

Hi-index 0.00

Visualization

Abstract

A spanning tree T of a graph G is said to be a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. While the complexity of the problem of recognizing whether a graph has a tree t-spanner is known for any fixed t≠3, the case t = 3 is still open. H.-O. Le and V. B. Le (1999, Networks, 34(2), 81-87) have shown that every directed path graph admits a tree 3-spanner by proposing an algorithm to construct a tree 3-spanner of a given directed path graph. In this paper, we point out a flaw in their algorithm by producing a directed path graph for which their algorithm fails to produce a tree 3-spanner although the graph admits a tree 3-spanner. Furthermore, we show that directed path graphs need not admit tree 3-spanners in general. Next, we show that directed path graphs of diameter two always admit tree 2-spanners and hence tree 3-spanners. Finally, we show that a tree 2-spanner of a diameter two directed path graph can be constructed in linear time. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(3), 203–210 2007