Determination and Analysis of Domination Numbers for Boundary Graph and Boundary Neighbour Graph Using MATLAB

Vertex domination is a key concept in graph theory, essential for analyzing the structural properties of graphs. This study explores the use of vertex domination to determine the domination numbers for Two specific graph structures: boundary graphs and boundary Neighbor graphs. A graph's dominance number is the minimal collection of vertex in which each vertex is simultaneously in the prevailing set and Neighboring to one of its vertex. A boundary graph is created by adding a pendant border to a circle, while the boundary graph consists of k linearly arranged paths joined at a common vertex, known as the spine. This research aims to identify the larger for each of these structures and employs MATLAB to implement the theoretical algorithms computationally. A MATLAB-based algorithm is provided to compute the domination number γ(H), identify the dominating set, and visualize the graph with highlighted dominated vertices. Computer experiments are conducted to test and validate the theoretical findings. This paper demonstrates the practical application of domination numbers using MATLAB, offering insights into the efficiency and structural properties of these graphs in both theoretical and real-world contexts.