DISCRETE LABELING IN GRAPHS AND DECOMPOSITION OF GRAPHS AND GRAPH-BASED OPTIMIZATION FRAMEWORK FOR WATER RESOURCE ALLOCATION

Abstract

In this paper we introduce a new type of labeling called Discrete Labeling with respect to decomposition of graphs. This labeling aims at providing the maximum output whenever the inputs are distinct with rational distribution of neighbours thereby exhibiting the stronger form of Cordial labeling. This work serves as an aid in decision making. The discrete labels 0 and 1 are cordially assigned to the vertices such that the edges receive labels depending on the incident vertex labels using EX-OR operation with the condition that for every vertex the cardinality of neighbours labeled 0 and 1 differs by at most 1. This paper proposes the discrete labeling of some standard and special graphs. Also it provides the cases for which certain path, star and cycle related graphs admit discrete labeling and also with respect to the deletion of vertices and edges.   The study also presents a novel Graph-Based Optimization Framework GBOF designed to address the complex task of sustainable water resource allocation in rural regions.

This labeling uses EX-OR operation which reduces the complexity of having twoswords in one sheath. Apart from keeping the distinct vertex labels and edge labels difference minimal, the cardinality of the neighbouring labels of every vertex are also taken into account.