# On vertex types of graphs

## TL;DR

This paper investigates the classification of vertices in graphs into seven types, addressing questions about the minimal size of graphs with specific distributions of these vertex types and providing answers to these classification questions.

## Contribution

The paper provides the first solutions to the open questions about the minimal order of graphs with many typical or pantypical vertices as posed by Hedetniemi et al.

## Key findings

- Determined the smallest order of graphs with n-2 very typical vertices.
- Determined the smallest order of graphs with n-2 typical vertices.
- Identified the minimal size of pantypical graphs.

## Abstract

The vertices of a graph are classified into seven types by J.T. Hedetniemi, S.M. Hedetniemi, S.T. Hedetniemi and T.M. Lewis and they ask the following questions: 1) What is the smallest order $n$ of a graph having $n-2$ very typical vertices or $n-2$ typical vertices? 2) What is the smallest order of a pantypical graph? We answer these two questions in this paper.