Let v(G) and p(G) be the number of vertices and the maximum number of
disjoint 3-vertex paths in G, respectively. We discuss the following old
Problem: Is the following claim (P) true ?
  (P) if G is a 3-connected and cubic graph, then p(G) = [v(G)/3], where
[v(G)/3] is the floor of v(G)/3.
  We show, in particular, that claim (P) is equivalent to some seemingly
stronger claims. It follows that if claim (P) is true, then Reed's dominating
graph conjecture (see [14]) is true for cubic 3-connected graphs.Comment: 24 pages and 11 figure