The hub number of co-comparability graphs

set H V is a hub set of a graph G = ( V , E ) if, for every pair of vertices u , v V H , either u is adjacent to v or there exists a path from u to v such that all intermediate vertices are in H. The hub number of G, denoted by h ( G ) , is the minimum size of a hub set in G. The connected hub number of G, denoted by h c ( G ) , is the minimum size of a connected hub set in G. In this paper, we prove that h ( G ) = h c ( G ) for co-comparability graphs G and characterize the case for which γ c ( G ) = h c ( G ) in this class of graphs, where γ c ( G ) denotes the connected domination number of G. We also show that h ( G ) can be computed in O ( | V | ) time for trapezoid graphs and in O ( | V | 3 ) time for co-comparability graphs.