Set-Valued Graphs

Abstract

A {\it set-indexer} of a given graph $G = (V, E)$ is an assignment $f$ of distinct nonempty subsets of a finite nonempty 'ground set' $X$ of cardinality $n$ to the vertices of $G$ so that the values $f^{\oplus}(e), e=uv \in E,$ obtained as the symmetric differences $f(u) \oplus f(v)$ of the subsets $f(u)$ and $f(v)$ of $X,$ are all distinct. A set-indexer $f$ of a graph $G,$ is called a {\it segregation} of $X$ on $G$ if the sets $f(V(G)) = \{f(u): u \in V(G)\}$ and $f^{\oplus}(E(G)) = \{f^{\oplus}(e): e \in E(G)\}$ are disjoint, and if, in addition, their union is the set $Y(X) = \mathcal{P}(X)-\{\emptyset\}$ of all the nonempty subsets of $X$ where $\mathcal{P}(X)$ denotes the power set of $X,$ then $f$ is called a {\it set-sequential labeling} of $G.$ A graph is hence called {\it set-sequential} if it admits a set-sequential labeling with respect to some 'ground set' $X.$ A set-indexer $f$ of a $(p, q)$-graph $G = (V, E)$ is called a {\it set-graceful labeling} of $G$ if there exists nonempty ground set $X$ such that $f^{\oplus}(E) = \mathcal{P}(X)-\{\emptyset\}$ and $G$ is {\it set-graceful} if it admits a set-graceful labeling. In this report we provide a complete characterization of set-sequential caterpillar of diameter four. We also a provide a new necessary condition for a graph to be set-sequential.