#### Title

Vertex Identification in Graphs

#### Date of Award

4-2022

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Ping Zhang, Ph.D.

#### Second Advisor

Gary Chartrand, Ph.D.

#### Third Advisor

John Martino, Ph.D.

#### Fourth Advisor

Dinesh Sarvate, Ph.D.

#### Keywords

Combinatorics, discrete mathematics, distance, graph theory, mathematics, vertex identification

#### Abstract

In recent decades, there has been increased interest in studying ways touniquely identify the vertices of a given graph. Two of best-known vertex identification methods are referred to as the metric dimension introduced by Slater and by Harary, and Melter in the1970s, and the partition dimension introduced by Chartrand in 2000. The first method deals with locating an ordered set of vertices in a connected graph G and uniquely identifying each vertex of G by means of the distance between this vertex and the vertices in the ordered set. The second method deals with partitioning the vertex set of a connected graph G into color classes and uniquely identifying each vertex of G by means of the distance between this vertex and a nearest vertex in each color class. The goal is either to minimize the size of such an ordered set or to minimize the number of colors needed. These two concepts have diverse real-life applications such as internet cyber security and developing the capability of large datasets of chemical graphs for pharmaceutical companies.

In this research, we study a different vertex identification method introduced by Chartrand in 2020. In this procedure, we seek an identification red-white coloring (an ID-coloring) of a given connected graph G by means of distance in a more general setting. An ID-coloring, should it exist, gives rise to distinct vectors, called codes, for the vertices of G, thereby uniquely identifying the vertices of G. Graphs possessing an ID-coloring are referred to as ID-graphs. We investigate the existence of ID-colorings in graphs, study structural and extremal problems dealing with ID-graphs and explore relationships among ID-colorings and traditional concepts in connected graphs. Results and open questions are presented in this area of research.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Kono, Yuya, "Vertex Identification in Graphs" (2022). *Dissertations*. 3819.

https://scholarworks.wmich.edu/dissertations/3819