Universal Minimal Resolving Functions in Graphs


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Authors

  • Varughese Mathew Department of Mathematics, Mar Thoma College Tiruvalla, Kerala, India
  • S. Arumugam National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH), Kalasalingam University, Tamil Nadu, India

Keywords:

Metric dimension, Fractional metric dimension, Resolving set, Resolving function, Universal minimal resolving function

Abstract

A vertex $x$ in a connected graph $G=(V,E)$ is said to resolve a pair $\{u,v\}$ of vertices of $G$ if the distance from $u$ to $x$ is not equal to the distance from $v$ to $x$. For the pair $\{u,v\}$ of vertices of $G$ the collection of all resolving vertices is denoted by $R\{u,v\}$ and is called the resolving neighborhood for the pair $\{u,v\}$. A real valued function $g : V \rightarrow [0,1]$ is a resolving function $(RF)$ of $G$ if $g(R\{u,v\}) \geq 1$ for all distinct pair $u,v \in V$. A resolving function $g$ is minimal ($MRF$) if any function $f:V\rightarrow [0,1]$ such that $f \leq g$ and $f(v) \neq g(v)$ for at least one $v \in V$ is nota resolving function of $G.$ A minimal resolving function $(MRF)$ is called a universal minimal resolving function $(UMRF)$ if its convex combination with every other $MRF$ is again an $MRF$. Minimal resolving functions are related to the fractional metric dimension of graphs. In this paper, we initiate a study of universal minimal resolving functions of a connected graph $G$.

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Published

15-03-2020

How to Cite

Varughese Mathew, & S. Arumugam. (2020). Universal Minimal Resolving Functions in Graphs. International Journal of Mathematics And Its Applications, 8(1), 231–237. Retrieved from http://ijmaa.in/index.php/ijmaa/article/view/195

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Section

Research Article