Set-Valued Graphs

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.