Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# An Algorithm for Coloring of Picture Fuzzy Graphs Based on Strong and Weak Adjacencies and its Application

Version 1 : Received: 6 November 2023 / Approved: 7 November 2023 / Online: 7 November 2023 (13:41:10 CET)

A peer-reviewed article of this Preprint also exists.

Rosyida, I.; Indrati, C.R. An Algorithm for Coloring of Picture Fuzzy Graphs Based on Strong and Weak Adjacencies, and Its Application. Algorithms 2023, 16, 551. Rosyida, I.; Indrati, C.R. An Algorithm for Coloring of Picture Fuzzy Graphs Based on Strong and Weak Adjacencies, and Its Application. Algorithms 2023, 16, 551.

## Abstract

The idea of strong and weak adjacencies between vertices has been generalized into fuzzy graphs and intuitionistic fuzzy graphs (IFGs) and it is an important part of making decisions. However, one or two membership degrees are not always sufficient for making decisions on real-world problems that need an answer of types “yes, neutral, and no". Consequently, in previous work, we generalized the concept into picture fuzzy graphs (PFGs) where each element in the PFG has membership, neutral, and non-membership degrees. Moreover, we constructed the notion of the coloring of PFGs based on strong and weak adjacencies between vertices. In this paper, we investigate some properties of the chromatic number of PFGs based on the concept of strong and weak adjacencies between vertices. According to the properties, we construct an algorithm to find the chromatic number of PFGs. The algorithm is usefull when we work with large PFGs. Further, we improve the method to implement the PFG’s coloring for determining traffic signal phasing at an intersection. A case study has also been done to evaluate the method.

## Keywords

Coloring; Chromatic Number; Picture Fuzzy Graph; Strong; Weak; Traffic Signal Phasing.

## Subject

Computer Science and Mathematics, Computational Mathematics