Set-Valued Graphs II

A {\it set-indexer} of a graph $G$ is an assignment of distinct subsets of a finite set $X_n$ 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 {\it set-sequential} if sets on the vertices and edges are distinct and together form the set of all nonempty subsets of $X_n.$ A set-indexer called {\it set-graceful} if all the nonempty subsets of $X_n$ are obtained on the edges. A graph is called {\it set-sequential} ({\it set-graceful}) if it admits a {\it set-sequential} ({\it set-graceful}) set-indexer. In the recent literature the notion of {\it set-indexer} has appeared as {\it 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.