How is an adjacency list implemented in a graph?

An adjacency list in a graph is implemented by storing a list of vertices connected to each vertex in the graph.

In more detail, an adjacency list is a data structure used to represent a graph in computer science. It is a way of defining the relationships between nodes (or vertices) in a graph. Each vertex in the graph has a list associated with it, which contains all the vertices that it is connected to. This list is known as the adjacency list of that vertex.

The adjacency list is usually implemented as an array or a linked list. Each element of the array or node of the linked list represents a vertex in the graph. The value stored in each element or node is a list of the vertices that are adjacent to the vertex represented by that element or node. This list can be implemented as a dynamic array, a linked list, a set, or any other collection data structure.

For example, consider a graph with four vertices labelled 1, 2, 3, and 4. If vertex 1 is connected to vertices 2 and 3, vertex 2 is connected to vertices 1 and 4, vertex 3 is connected to vertices 1 and 4, and vertex 4 is connected to vertices 2 and 3, the adjacency list representation of this graph would be as follows:

1: 2, 3
2: 1, 4
3: 1, 4
4: 2, 3

This means that the adjacency list for vertex 1 contains vertices 2 and 3, the adjacency list for vertex 2 contains vertices 1 and 4, and so on.

The adjacency list representation of a graph is more space-efficient than the adjacency matrix representation, especially for sparse graphs (i.e., graphs with few edges), because it only stores the connections that actually exist. However, it can be slower to use than the adjacency matrix representation for certain operations, such as checking whether an edge exists between two vertices.