Domination Index in Graphs

TL;DR

This paper introduces the domination index in graphs based on domination degree, explores its properties across various graph types, and discusses algorithms and applications in facility allocation.

Contribution

It presents a new concept of domination index using domination degree, extending its analysis to various graph classes and operations, and proposes algorithms for minimal dominating set identification.

Findings

Domination index defined and analyzed for multiple graph types.

Inequalities involving domination degree and other graph parameters established.

Algorithm for finding a minimal dominating set containing a specific vertex developed.

Abstract

The concepts of domination and topological index hold great significance within the realm of graph theory. Therefore, it is pertinent to merge these concepts to derive the domination index of a graph. A novel concept of the domination index is introduced, which utilizes the domination degree of a vertex. The domination degree of a vertex a is defined as the minimum cardinality of a minimal dominating set that includes a. The idea of domination degree and domination index is conducted of graphs like complete graphs, complete bipartite, r partite graphs, cycles, wheels, paths, book graphs, windmill graphs, Kragujevac trees. The study is extended to operation in graphs. Inequalities involving domination degree and already established graph parameters are discussed. An application of domination degree is discussed in facility allocation in a city. Algorithm to find a MDS containing a particular vertex is also discussed in the study.