Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

# Vertex Domination in Fuzzy Graphs

Version 1 : Received: 3 April 2018 / Approved: 8 April 2018 / Online: 8 April 2018 (08:24:36 CEST)
Version 2 : Received: 28 April 2019 / Approved: 29 April 2019 / Online: 29 April 2019 (12:34:08 CEST)
Version 3 : Received: 29 April 2019 / Approved: 5 May 2019 / Online: 5 May 2019 (12:20:38 CEST)

How to cite: Nikfar, M. Vertex Domination in Fuzzy Graphs. Preprints 2018, 2018040085 (doi: 10.20944/preprints201804.0085.v3). Nikfar, M. Vertex Domination in Fuzzy Graphs. Preprints 2018, 2018040085 (doi: 10.20944/preprints201804.0085.v3).

## Abstract

We introduce a new variation on the domination theme which we call vertex domination as reducing waste of time in transportation planning and optimization of transport routes. We determine the vertex domination number $\gamma_v$ for several classes of fuzzy graphs. The bounds is obtained for it. In fuzzy graphs, monotone decreasing property and monotone increasing property are introduced. We prove both of the vizing's conjecture and the Grarier-Khelladi's conjecture are monotone decreasing fuzzy graph property for vertex domination. We obtain Nordhaus-Gaddum (NG) type results for these parameters. The relationship between several classes of operations on fuzzy graphs with the vertex domination number of them is studied. Finally, we discuss about vertex dominating set of a fuzzy tree by using the bridges and $\alpha$-strong edges equivalence.

## Subject Areas

fuzzy graph; fuzzy bridge; fuzzy tree; $\alpha$-strong arc; vertex domination

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