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Set-Valuations of Graphs and Their Applications

Set-Valuations of Graphs and Their Applications
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Author(s): Germina K. Augusthy (Central University of Kerala, India)
Copyright: 2020
Pages: 37
Source title: Handbook of Research on Advanced Applications of Graph Theory in Modern Society
Source Author(s)/Editor(s): Madhumangal Pal (Vidyasagar University, India), Sovan Samanta (Tamralipta Mahavidyalaya, India) and Anita Pal (National Institute of Technology Durgapur, India)
DOI: 10.4018/978-1-5225-9380-5.ch008

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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.

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