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SetValuations of Graphs and Their Applications

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: 10.4018/9781522593805.ch008
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Abstract
A setvaluation 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 setindexer of G is an injective setvaluation f:V(G)→2x such that the induced setvaluation 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. Setvalued graphs such as setgraceful graphs, topological setgraceful graphs, setsequential graphs, setmagic graphs are discussed. Setvaluations with a metric, associated with each pair of vertices is defined as distance pattern distinguishing (DPD) set of a graph (opendistance 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 distancepattern 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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