A Λ - factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v ( G ) be the number of vertices of G and γ ( G ) the domination number of G . A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it does not have an induced subgraph isomorphic to a claw. Our results include the following. Let G be a 3-connected claw-free graph, x ∈ V ( G ) , e = x y ∈ E ( G ) , and L a 3-vertex path in G . Then ( a 1 ) if v ( G ) ≡ 0 mod 3 , then G has a Λ -factor containing (avoiding) e , ( a 2 ) if v ( G ) ≡ 1 mod 3 , then G − x has a Λ -factor, ( a 3 ) if v ( G ) ≡ 2 mod 3 , then G − { x , y } has a Λ -factor, ( a 4 ) if v ( G ) ≡ 0 mod 3 and G is either cubic or 4-connected, then G − L has a Λ -factor, ( a 5 ) if G is cubic with v ( G ) ≥ 6 and E is a set of three edges in G , then G − E has a Λ -factor if and only if the subgraph induced by E in G is not a claw and not a triangle, ( a 6 ) if v ( G ) ≡ 1 mod 3 , then G − { v , e } has a Λ -factor for every vertex v and every edge e in G , ( a 7 ) if v ( G ) ≡ 1 mod 3 , then there exist a 4-vertex path Π and a claw Y in G such that G − Π and G − Y have Λ -factors, and ( a 8 ) γ ( G ) ≤ ⌈ v ( G ) / 3 ⌉ and if in addition G is not a cycle and v ( G ) ≡ 1 mod 3 , then γ ( G ) ≤ ⌊ v ( G ) / 3 ⌋ . We also explore the relations between packing problems of a graph and its line graph to obtain some results on different types of packings and discuss relations between Λ -packing and domination problems.