Radius versus diameter in cocomparability and intersection graphs (Q1356529)

Let \(r\) be the radius and \(d\) the diameter of a given graph. Then evidently \(r\leq d \leq 2r\) and this has been shown to be sharp in general. For a cocomparability graph of a partially ordered set (with edges between all incomparable pairs) one has \(2r \leq d+3\). The intersection graph of a finite family of sets has edges between all nondisjoint pairs. For a trapezoid graph, i.e. the intersection graph of a family a trapezes between two fixed parallel lines, \(2r \leq d+2\) holds. For a (proper) circular-arc graph, i.e. the intersection graph of a (pairwise incomparable) set of arcs of a fixed circle, we have \(d\leq r+2\) (\(d\leq r+1\)).