Some graceful lobsters with all three types of branches incident on the vertices of the central path

We observe that a lobster having diameter at least five has a unique path H=x0x1…xm satisfying the property that, besides the adjacencies in HH, both x0x0 and xmxm are adjacent to the centers of at least one K1,sK1,s, where s>0s>0, and each xixi, for 1≤i≤m−11≤i≤m−1, is at most adjacent to the centers of some K1,sK1,s, where s≥0s≥0. This unique path HH is called the central path   of the lobster. We call K1,sK1,s an even branch   if ss is even and s≠0s≠0, an odd branch   if ss is odd, and a pendant branch   if s=0s=0. In this paper, we give graceful labelings to the lobsters having one of the following features. Let l1l1, l2l2, and l3l3 be integers such that 1≤l1