In an adjacency list implementation, we have an array of |V| vertices. Within each array cell, we place a list containing all of that vertex's neighbors.

0 -> {1}
1 -> {1,3,4,5}
2 -> null
3 -> null
4 -> null
5 -> {1,6}
6 -> null

In this implementation, we can see how easy it is to add vertices and remove them. In a sparse graph, the efficiency is on average O(1).

The space it takes it O(E+V), much less than adjacency matrix implementation.

Adding a vertex is simple. Just append a new vertex containing an empty list to the end of our ArrayList.

int v = getNumVertices();
ArrayList<Integer> neighbors = new ArrayList<Integer>();
setNumVertices(v+1);
}

## Deleting a Vertex

To deleting a vertex, we remove the last ArrayList in our Map.

public void removeVertex() throws VertexOutOfBoundsException {
// Remove the vertex at the end
int numV = getNumVertices();
if (numV == 0)
throw new VertexOutOfBoundsException();
setNumVertices(numV+1);
}

To add an edge, simply retrieve the ArrayList corresponding to the beginning vertex in our Map, then append the value of the end vertex.

public void addEdge(int v, int w) throws VertexOutOfBoundsException {
int numV = getNumVertices();
if (numV <= v || numV <= w) {
throw new VertexOutOfBoundsException();
}
setNumEdges(getNumEdges()+1);
}

## Removing an Edge

To remove an edge that starts from v and goes to w, remove it from the vertex's list.

public void removeEdge(int v, int w) throws VertexOutOfBoundsException {
int numV = getNumVertices();
if (numV <= v || numV <= w) throw new VertexOutOfBoundsException();
// Remove edge that starts from v to w
setNumEdges(getNumEdges()+1);
}

## Finding in/out-degree

To find the in-degree, find the size of the corresponding vertex in the adjacency list. For out-degree, we must traverse through all ArrayLists in the entire adjacency list and count the number of times our vertex appears.

public int getInDegree(int v) throws VertexOutOfBoundsException {
int numV = getNumVertices();
if (numV <= v) throw new VertexOutOfBoundsException();
int count = 0;
for (int i = 0; i < getNumVertices(); i++) {
count++;
}
}
return count;
}

public int getOutDegree(int v) throws VertexOutOfBoundsException {
int numV = getNumVertices();
if (numV <= v) throw new VertexOutOfBoundsException();
}

## Finding degree sequence

The degree sequence is the same as the Adjacency Matrix implementation.

public List<Integer> getDegreeSeq() throws VertexOutOfBoundsException {
List<Integer> degreeSeq = new ArrayList<Integer>();
int degrees = 0;
for (int i = 0; i < getNumVertices(); i++) {
degrees = getInDegree(i) + getOutDegree(i);
}
Collections.sort(degreeSeq);
Collections.reverse(degreeSeq);
return degreeSeq;
}

To obtain a list of all adjacent neighbors, simply return a copy of the list stored in our List.

public List<Integer> getNeighbors(int v) throws VertexOutOfBoundsException {
int numV = getNumVertices();
if (numV <= v) throw new VertexOutOfBoundsException();

List<Integer> neighbors = new ArrayList<Integer>();
for (Integer x : adjListsMap.get(v)) {
}

return neighbors;
}

## Getting all two-distanced neighbors

For getting all two-distanced neighbors, find all one-distanced neighbors, then find the neighbors of those.

public List<Integer> getNeighborsTwoApart(int v) throws VertexOutOfBoundsException {
int numV = getNumVertices();
if (numV <= v) throw new VertexOutOfBoundsException();

List<Integer> oneApart = getNeighbors(v);
ArrayList<Integer> twoApart = new ArrayList<Integer>();
// For each integer within one hop of v...
for (int i = 0; i < oneApart.size(); i++) {
for (Integer x : oneApart) {
}
}
return twoApart;
}

Calculate the efficiencies of these operations and compare them to our Adjacency Matrix implementation!

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