  A subgraph (a spanning subgraph) of a graph G whose all components are
3-vertex paths is called an L-packing (respectively, an L-factor} of G. We
discuss the following old
  PROBLEM (A. Kelmans, 1984). Is the following claim true?
  (C) If G is a cubic 3-connected graph, then G has an L-packing that avoids at
most two vertices of G.
  We show, in particular, that claim (C) is equivalent to some seemingly
stronger claims (see Theorem 3.1 below).
  For example, if G is a cubic 3-connected graph and the number of vertices of
G is divisible by three, then then the following claims are equivalent: G has
an L-factor, for every edge e of G there is an L-factor of G avoiding
(containing) e, G - {e,f} has an L-factor for every two edges e and f of G, and
G - P has an L-factor for every 3-vertex path P in G.
  It follows that if claim (C) is true, then Reed's dominating graph conjecture
is true for cubic 3-connected graphs.
  We also show that certain claims in Theorem 3.1 are best possible. We give a
construction providing infinitely many cyclically 6-connected graphs G with two
disjoint 3-vertex paths P and P' such that the number of vertices of G is
divisible by three and G - P- P' has no L-factor.
  Keywords: cubic 3-connected graph, 3-vertex path packing, 3-vertex path
factor, domination.
