Preprint Article Version 2 This version is not peer-reviewed

The Results on 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. The Results on Vertex Domination in Fuzzy Graphs. Preprints 2018, 2018040085 (doi: 10.20944/preprints201804.0085.v2). Nikfar, M. The Results on Vertex Domination in Fuzzy Graphs. Preprints 2018, 2018040085 (doi: 10.20944/preprints201804.0085.v2).

Abstract

We do fuzzification the concept of domination in crisp graph by using membership values of nodes, $\alpha$-strong arcs and arcs. In this paper, we introduce a new variation on the domination theme which we call vertex domination. We determine the vertex domination number $\gamma_v$ for several classes of fuzzy graphs, specially complete fuzzy graph and complete bipartite fuzzy graphs. The bounds is obtained for the vertex domination number of fuzzy graphs. Also the relationship between $M$-strong arcs and $\alpha$-strong is obtained. In fuzzy graphs, monotone decreasing property and monotone increasing property is introduced. We prove the vizing's conjecture is monotone decreasing fuzzy graph property for vertex domination. we prove also the Grarier-Khelladi's conjecture is monotone decreasing fuzzy graph property for it. 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.

Subject Areas

fuzzy graph; $\alpha$-strong arcs; weight of nodes; vertex domination