A dominating set of a graph G is a set D subset of V (G) such that every vertex in V (G)\D has a neighbor in D, where two vertices are neighbors if they are adjacent. The domination number of G, denoted by gamma(G), is the minimum cardinality among all dominating sets of G. A packing of a graph G is a set of vertices that are mutually distance at least 3 apart. The packing number of G, denoted by rho(G), is the maximum cardinality among all packings of G. It is conjectured that gamma(G) <= 2 rho(G) if G is a connected graph with maximum degree at most 3, except for three graphs. It has also been shown that if G is a claw-free graph with maximum degree at most 3, then gamma(G) <= 2 rho(G). In this paper, we show that gamma(G) <= 2 rho(G) if G is a connected P-5-free graph with maximum degree at most 3. We further show that if G is a connected H-free graph for some H is an element of {P-3 boolean OR P-2, 2K(2) boolean OR K-1} with maximum degree at most 3 except for some finite set of graphs, then gamma(G) <= 2 rho(G).
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