Some Results on Generating Graceful Trees - Venkatesh, S.; Balasubramanian, K. (International Journal of Engineering and Technology, 2018)
Let  and  be any two simple graphs. Then  is the graph obtained by merging a vertex of each copy of  with every attachment vertices of . Let  be the one vertex union of copies of the given caterpillar  with the common vertex as one of the penultimate vertices. If  is any caterpillar, then define . Recursively for , construct ,that is,   Here the tree  considered for attachment with  is a caterpillar, but not necessarily the same among the levels. In this paper we prove that the tree  is graceful for