The hub number of a graph

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) hc(G) c(G) h(G) + 1, where h(G), hc(G), and 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 c(G) > hc(G) 4 are obtained by substituting graphs into three consecutive vertices of a cycle; this yields a polynomial-time algorithm to check whether hc(G) = c(G).

• Type: Journal
• Year: 2008
• Where: IPL
• Authors: Tracy Grauman, Stephen G. Hartke, Adam Jobson, Bill Kinnersley, Douglas B. West, Lesley Wiglesworth, Pratik Worah, Hehui Wu
• Keywords: Connected Subgraph, Internal Vertices, IPL 2008, Minimum Sizes