A set-indexer of a graph G is an assignment of distinct subsets of a finite set of n elements to the vertices of the graph, where the edge values are obtained as the symmetric differences of the set assigned to their end vertices which are also distinct. A set-indexer is called set-sequential if sets on the vertices and edges are distinct and together form the set of all nonempty subsets of A set-indexer called set-graceful if all the nonempty subsets of are obtained on the edges. A graph is called set-sequential (set-graceful) if it admits a set-sequential (set-graceful) set-indexer. In the recent literature the notion of set-indexer has appeared as set-coloring. While obtaining in general a `good' characterization of a set-sequential (set-graceful) graphs remains a formidable open problem ever since the notion was introduced by Acharya in 1983, it becomes imperative to recognize graphs which are set-sequential (set-graceful). In particular, the problem of characterizing set-sequential trees was raised raised by Acharya in 2010. In this article we completely characterize the set-sequential caterpillars of diameter five.
K. Abhishek, “Set-Valued Graphs II”, Journal of Fuzzy Set Valued Analysis, vol. 2013 , p. 16, 2013.