Abstract
A 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 hc(G), is the minimum size of a connected hub set in G. In this paper, we prove that h(G)=hc(G) for co-comparability graphs G and characterize the case for which γc(G)=hc(G) in this class of graphs, where γc(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.
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