The connected hub number and the connected domination number

The connected hub number hc(G) of a connected graph G is the smallest order of a connected subgraph H of G such that any two nonadjacent vertices of G − H are joined in G by a path with all internal vertices in H. Letting γc(G) denote the connected domination number of G, it is easy to see that hc(G) ≤ γc(G) ≤ hc(G) + 1 for every connected graph G. Here we characterize the graphs G for which γc(G) = hc(G) + 1. Our result contributes to the search for the solution of an extremal problem of (Newman-Wolfe et al., Congressus Numerantium 67 (1988), 67–76). © 2011 Wiley Periodicals, Inc. NETWORKS, 2011 © 2011 Wiley Periodicals, Inc.