Set-Valuations of Graphs and Their Applications

Abstract

A set-valuation of a graph G=(V,E) assigns to the vertices or edges of G elements of the power set of a given nonempty set X subject to certain conditions. A set-indexer of G is an injective set-valuation f:V(G)→2x such that the induced set-valuation f⊕:E(G)→2X on the edges of G defined by f⊕(uv)=f(u)⊕f(v) ∀uv∈E(G) is also injective, where ⊕ denotes the symmetric difference of the subsets of X. Set-valued graphs such as set-graceful graphs, topological set-graceful graphs, set-sequential graphs, set-magic graphs are discussed. Set-valuations with a metric, associated with each pair of vertices is defined as distance pattern distinguishing (DPD) set of a graph (open-distance pattern distinguishing set of a graph (ODPU)) is ∅≠M⊆V(G) and for each u∈V(G), fM(u)={d(u,v): v ϵ M} be the distance-pattern of u with respect to the marker set M. If fM is injective (uniform) then the set M is a DPD (ODPU) set of G and G is a DPD (ODPU)-graph. This chapter briefly reports the existing results, new challenges, open problems, and conjectures that are abound in this topic.