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A subgraph (a spanning subgraph) of a graph G whose all components are3-vertex paths is called an L-packing (respectively, an L-factor} of G. Wediscuss 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 atmost two vertices of G.  We show, in particular, that claim (C) is equivalent to some seeminglystronger claims (see Theorem 3.1 below).  For example, if G is a cubic 3-connected graph and the number of vertices ofG is divisible by three, then then the following claims are equivalent: G hasan 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, andG - 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 conjectureis true for cubic 3-connected graphs.  We also show that certain claims in Theorem 3.1 are best possible. We give aconstruction providing infinitely many cyclically 6-connected graphs G with twodisjoint 3-vertex paths P and P' such that the number of vertices of G isdivisible by three and G - P- P' has no L-factor.  Keywords: cubic 3-connected graph, 3-vertex path packing, 3-vertex pathfactor, domination.

Packing 3-vertex Paths In Cubic 3-connected Graphs