Set-Valued Graphs II

A set-indexer of a graph is an assignment of distinct subsets of a finite set of 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 `goodʹ 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.