Graceful lobsters obtained by partitioning and component moving of branches of diameter four trees

For any path P of maximum length in a lobster of diameter at least five, the path obtained from P by deleting two vertices from both the ends is called the central path of the lobster. Wang et al. [1] have given graceful labelings to lobsters in which all the vertices except one end vertex, say x0, of the central path are attached to the centers of a nonzero even number of stars K1,s, s ≥ 1, and x0 is attached to the centers of an odd number (≥ 3) of stars K1,s, s ≥ 1. Moreover, in all these lobsters the stars K1,s are of one type, i.e., either s is odd or even throughout. In this paper, we give graceful labelings to some new classes of lobsters in which the vertices of the central path have the same property as above except for one important feature, that the stars K1,s, s ≥ 0, (notice that we also consider the case s = 0) incident on the central path need not be of one type.