Sharp Bounds for the Oriented Diameters of Interval Graphs and 2-Connected Proper Interval Graphs

  • Authors:

  • Jing Huang;Dong Ye

  • Affiliations:

  • Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria, B.C., Canada, V8W 3P4, ;School of mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, People's Republic of China

  • Venue:
  • ICCS '07 Proceedings of the 7th international conference on Computational Science, Part III: ICCS 2007
  • Year:
  • 2007

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Abstract

The diameter diam(H) of a (directed) graph H is the maximum value computed over the lengths of all shortest (directed) paths in H. Given a bridgeless connected graph G, the oriented diameter OD(G) is given by $OD(G) = \mbox{min}\{diam(H):\ H\ \mbox{is\ an\ orientation\ of}\ G\}$. In this paper, we show that $OD(G) \leq \lceil \frac{3}{2} diam(G) \rceil + 1$ for every connected bridgeless interval graph G. Further, for every 2-connected proper interval graph G, we show $OD(G) \leq \lceil \frac{5}{4} diam(G) \rceil + 1$ if diam(G) ≤ 3 and $OD(G) \leq \lceil \frac{5}{4} diam(G) \rceil$, otherwise. All the bounds are sharp and improve earlier estimations obtained by Fomin et al.