In this post we are going to talk about the graph data structure. A graph is a data structure that consists of vertices (also known as nodes) and a set of edges that connect the vertices. It is used to represent relationships between objects such as one’s social network friends, cities in a country or build dependencies in a build pipeline. Before we jump into the code we want to look into the theory of graphs – different types of graphs and different ways to represent them.

## Directed Vs Undirected Graphs

A graph can be either directed or undirected. A directed graph (also known as a digraph) is a graph in which the edges have a direction. In Figure 1 below there is an edge between 1 and 3 with a direction going towards 3. This means you can go from 1 to 3 but not the other way round. Think of it like a one-way road from 1 to 3.

An undirected graph is a graph in which edges don’t have specific directions. You can move from one vertex to the other and vice versa. Figure 2 below is an example of an undirected graph. You can move from 1 to 3 as well as from 3 to 1.

## Weighted Vs Unweighted Graph

A weighted graph is a graph in which the edges have a numerical weight associated with them. Using an example of a graph representing connections between cities the weights can represent the distance between two cities. Figure 3 below is an example of a weighted graph while figures 1 and 2 above are examples of unweighted graphs.

## Representing Graphs In Code

There are two ways of representing a graph in code – adjacency matrix and adjacency list. Each one has its pros and cons.

An adjacency matrix is a $$n \times n$$ matrix where $$n$$ is the number of vertices in a graph. In code we can use a two dimensional array. When there is an edge between two vertices $$i$$ and $$j$$ the value at $$A[ i ][ j ]$$ is set to 1 (true) otherwise the value is 0 (false). Figure 4 below is an example of an adjacency matrix representing the undirected graph in Figure 2 above.

Adjacency matrices are very useful when you want to quickly check if there is a connection between two nodes. You just get the value at $$A[ i ][ j ]$$ in $$O(1)$$ time. We can also add or delete an edge in $$O(1)$$ time. However, adjacency matrix take too much space – $$n^2$$ where $$n$$ is the number of vertices. If you have only one edge in that graph then you will have a lot of wasted space. Traversing an adjacency matrix requires $$O(n^2)$$ time since it’s a two dimensional array. I still have nightmares from my time in school when we were using adjacency matrices in my Graph Theory class. And to make it worse, we were doing everything by hand .

An adjacency list is a list of vertices with each vertex containing a list of vertices it’s connected to. There are multiple ways of representing this in code. I prefer using a dictionary (hash map) with the keys being the vertices and the values a set (hash set) of vertices. Something like Dictionary<T, HashSet<T>>. The reason why I like this representation is because adding or removing an edge takes only $$O(1)$$ time. Also checking is there is an edge between two vertices takes $$O(1)$$. Had we used an array then we would be looking at $$O(n)$$ time complexity. Figure 5 below is an adjacency list representation of the graph in Figure 2 above.

## The Code

We are going to create a basic graph using an adjacency list and add one function to add a vertex. Other functions are going to be added in subsequent posts.

public class HDGraph<T>
{

public HDGraph()
{
}

{
{