On domination in 2-connected cubic graphs

In 1996, Reed proved that the domination number, gamma(G), of every n-vertex graph G with minimum degree at least 3 is at most 3n/8 and conjectured that gamma(H) <= [n/3] for every connected 3-regular (cubic) n-vertex graph H. In[1] this conjecture was disproved by presenting a connected cubic graph G on 60 vertices with gamma(G) = 21 and a sequence {G(k)}(k=1)(infinity) of connected cubic graphs with lim(k ->infinity)gamma(G(k))/vertical bar V(G(k))vertical bar >= 1/3+1/69. All the counter-examples, however, had cut-edges. On the other hand, in[2] it was proved that gamma(G) <= 4n/11 for every connected cubic n-vertex graph G with at least 10 vertices. In this note we construct a sequence of graphs {G(k)}(k=1)(infinity) of 2-connected cubic graphs with lim(k ->infinity)gamma(G(k))/vertical bar V(G(k))vertical bar >= 1/3+1/78, and a sequence {G(l)'}(l=1)(infinity) of connected cubic graphs where for each G(l)' we have gamma(G(l)')/vertical bar V(G(l)')vertical bar > 1/3+1/69.