Complementary Topological Graphs: Structural Properties and Domination Parameters

Abstract

In this thesis, a new type of graph called the complementary topological graph, denoted by ​, is introduced and studied. This graph is constructed from a finite set  equipped with the discrete topology, where the vertices represent all non-empty proper subsets of , and two vertices are adjacent whenever their union equals the whole set  . The work focuses on studying several structural properties of , including the order, size, degree of vertices, Eulerian properties, and chromatic number. It is shown that the graph contains vertices and ​ edges, where . In addition, the graph is proved to be regular with degree . Based on these properties, it is concluded that ​ is neither Eulerian nor semi-Eulerian for all . The chromatic number of the graph is also determined and shown to be equal to .

Furthermore, this thesis investigates domination in complementary topological graphs under some graph operations, particularly the corona and join operations. The domination number for these operations is obtained and characterized through several results and examples. The results presented in this thesis provide a deeper understanding of the relationship between graph theory and topology, and they may serve as a basis for future studies on complementary topological graphs and their applications in different areas of graph theory.