The hub number of a graph

Abstract

A hub set in a graph G is a set U ⊆ V ( G ) such that any two vertices outside U are connected by a path whose internal vertices lie in U. We prove that h ( G ) ⩽ h c ( G ) ⩽ γ c ( G ) ⩽ h ( G ) + 1 , where h ( G ) , h c ( G ) , and γ c ( G ) , respectively, are the minimum sizes of a hub set in G, a hub set inducing a connected subgraph, and a connected dominating set. Furthermore, all graphs with γ c ( G ) > h c ( G ) ⩾ 4 are obtained by substituting graphs into three consecutive vertices of a cycle; this yields a polynomial-time algorithm to check whether h c ( G ) = γ c ( G ) .