We observe that a lobster having diameter at least five has a unique path satisfying the property that, besides the adjacencies in H, both x₀ and x_m are adjacent to the centers of at least one K_1,s, where s>0, and each x_i, for 1≤i≤m−1, is at most adjacent to the centers of some K_1,s, where s≥0. This unique path H is called the central path of the lobster. We call K_1,s an even branch if s is even and s≠0, an odd branch if s is odd, and a pendant branch if s=0. In this paper, we give graceful labelings to the lobsters having one of the following features. Let l₁, l₂, and l₃ be integers such that 1≤l₁<l₂≤l₃≤m.

(1) The vertex x₀ is attached to an even number of odd branches and an odd number of pendant branches. Each vertex x_i, for 1≤i≤l₁, is attached to some combination consisting of odd and pendant branches, whereas each vertex x_i, for l₁+1≤i≤l₂, is attached to some combination consisting of all three types of branches. If l₂<m, then we have one of the following properties.

(i) Each vertex x_i, for l₂+1≤i≤l₃, is attached to some combination consisting of odd and even branches and each of the rest of the vertices x_i, if any, is attached to only odd (or even) branches.

(ii) Each vertex x_i, for l₂+1≤i≤l₃, is attached to some combination consisting of odd (respectively, even) and pendant branches. Each of the rest of the vertices x_i, if any, is attached to only odd (respectively, even) branches.

(2) The vertex x₀ is attached to an even number of odd branches and an odd number of pendant branches and combinations of branches incident on the vertices x_i, for 1≤i≤m, are the subcases of those mentioned in (1), where one or more combinations are absent.