  Let v(G) and dom(G) denote the number of vertices and the domination number
of a graph G, and let r (G) = dom(G)/v(G)$. Let [x] and ]x[ be the floor and
the ceiling of a number x. In 1996 B. Reed conjectured that if G is a cubic
graph, then dom(G) is at most ]v(G)/3[. In 2005 A. Kostochka and B. Stodolsky
disproved this conjecture for cubic graphs of connectivity one and maintained
that the conjecture may still be true for 2-connected cubic graphs. Their
minimum counterexample C has 4 bridges, v(C) = 60, anddom (C) = 21. In this
paper we disprove Reed's conjecture for 2-connected cubic graphs by providing a
sequence (R(k): k > 2) of cubic graphs of connectivity two with r(R_k) = 1/3 +
1/60, where v(R(k+1)) > v(R(k)) > v(R(3)) = 60 for k > 3, and so dom(R(3)) =
21$ and dom(R(k)) - ]v(R(k))/3[ tends to infinity when k tends to infinity. We
also provide a sequence of (L_s: s > 0) of cubic graphs of connectivity one
with r(L(s)) > 1/3 + 1/60. The minimum counterexample L = L(1) in this sequence
is `better' than C in the sense that L has 2 bridges while C has 4 bridges,
v(L) = 54 < 60 = v(C), and r(L) = 1/3 + 1/54} > 1/3 + 1/60 = r(C). We also give
a construction providing for every t in {0,1,2} infinitely many cubic
cyclically 4-connected Hamiltonian graphs G(t) such that v(G(t)) = t mod 3, t
in {0,2} implies dom(G(t)) = ]v(G(r))/3[, and t = 1 implies dom(G(t)) =
[v(G(r))/3]. At last we suggest a stronger conjecture on domination in cubic
3-connected graphs.

Combinatorics