Set-Valuations of Graphs and their Applications: A Survey

A set-valuation of a graph G = (V, E) assigns to the vertices or edges of G elements of the power set X 2 of a given nonempty set X subject to certain conditions and set-valuations have a variety of origins. Acharya defined a set-indexer of G to be an injective set-valuation X G V f 2 ) ( : → such that the induced set-valuation ) ( : G E f ⊕ X 2 → on the edges of G defined by ) ( ), ( ) ( ) ( G E uv v f u f uv f ∈ ∀ + = ⊕ is also injective, where ⊕ denotes the operation of taking the symmetric difference of the subsets of X. In particular, he studied variety of set-valued graphs such as set-graceful graphs, topological set-graceful graphs, set-sequential graphs, set-magic graphs, etc. In 2006, Acharya and Germina defined the concept of distance pattern distinguishing set of a graph (open-distance pattern distinguishing set of a graph). Let G = (V, E) be a given connected simple (p, q)-graph with diameter , G d ∅ ≠ M ⊆ V(G) and for each u ∈ V(G), let fM(u )= {d(u,v): v ∈M} be the distance-pattern of u with respect to the marker set M. If M f is injective (uniform) then the set M is a DPD-set (ODPU-set) of G and G is a DPD-graph (ODPU-graph). Following a suggestion made by Michel Deza, Acharya and Germina, who had been studying topological set-valuations, introduced the particular kind of set-valuations for which a metric, especially the cardinality of the symmetric difference, is associated with each pair of vertices in proportion to the distance between them in the graph. Particular cases of set-valuations of graphs are also being studied in detail by many authors. In this paper, we give a brief report of the existing results, new challenges, open problems and conjectures that are abound in this area of set-valuations of graphs.