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

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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.