Domination number versus packing number in graphs

A dominating set of a graph G is a set D ⊆ 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 γ(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 ρ(G), is the maximum cardinality among all packings of G.It is conjectured that γ(G) ≤ 2ρ(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 γ(G) ≤ 2ρ(G).In this paper, we show that γ(G) ≤ 2ρ(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 ∈ {P 3 ∪ P 2 , 2K 2 ∪ K 1 } with maximum degree at most 3 except for some finite set of graphs, then γ(G) ≤ 2ρ(G).