Undirected Graphs

Glossary

• A self-loop is an edge that connects a vertex to itself.

• Two edges are parallel if they connect the same pair of vertices.

• When an edge connects two vertices, we say that the vertices are adjacent to one another and that the edge is incident on both vertices.

• The degree of a vertex is the number of edges incident on it.

• A subgraph is a subset of a graph's edges (and associated vertices) that constitutes a graph.

• A path in a graph is a sequence of vertices connected by edges, with no repeated edges.

• A simple path is a path with no repeated vertices.

• A cycle is a path (with at least one edge) whose first and last vertices are the same.

• A simple cycle is a cycle with no repeated vertices (other than the requisite repetition of the first and last vertices).

• The length of a path or a cycle is its number of edges.

• We say that one vertex is connected to another if there exists a path that contains both of them.

• A graph is connected if there is a path from every vertex to every other vertex.

• A graph that is not connected consists of a set of connected components, which are maximal connected subgraphs.

• An acyclic graph is a graph with no cycles.

• A tree is an acyclic connected graph.

• A forest is a disjoint set of trees.

• A spanning tree of a connected graph is a subgraph that contains all of that graph's vertices and is a single tree. A spanning forest of a graph is the union of the spanning trees of its connected components.

• A bipartite graph is a graph whose vertices we can divide into two sets such that all edges connect a vertex in one set with a vertex in the other set.

Graph having two directions from any vertex in a path to another verted.

Representation of undirected Graph

Adjacency Matrix Representations

Algorithm

Maintain a two dimensional $v * v$ matrix boolean array

for each edge $v - w$ in graph: $adj[v][w] = adj[w][v] = true$

Adjacency Matrix representation

Note: Adjacency matrix consumes more space.

• Space Complexity : $v^{2}$

• Add Edge: 1

• Edge between $v$and $w$ is 1

• Iterate over vertices adjacent to $v$ : $V$

Adjacency List Representation

In this representation , If there is path exists between any two vertices, vertices are added to vertex list.

Java Implementation

public class Graph {
  private final int V;
  private List<Integer>[] adj;

  public Graph(int v) {
    this.V = v;
    adj = new List[v];
    for (int i = 0; i < V; i++) {
      adj[i] = new ArrayList<>();
    }
  }

  public void addEdge(int v, int w) {
    adj[v].add(w);
    adj[w].add(v);
  }

  public Iterable<Integer> adj(int v) {
    return adj[v];
  }

  public static void main(String[] args) {}
}

• Space Complexity : $E + V$

• Add Edge: 1

• Edge between $v$and $w$ is $degree(v)$

• Iterate over vertices adjacent to $v$ - $degree(v)$

Depth First Search

Algorithm

To visit a vertex $v$ :

• Mark vertex $v$ as visited.

• Recursively visit all unmarked vertices adjacent to $v$.

Java Implementation

public class DepthFirstPaths
{
   // marked[v] = true if v connected to s
   private boolean[] marked;
   
   // edgeTo[v] = previous vertex on path from s to v
   private int[] edgeTo;
   private int s;
   
   public DepthFirstPaths(Graph G, int s)
   {
      // Initialize all the constructors
      ..............
      
      dfs(G, s); 
   }
      
   private void dfs(Graph G, int v)
   {
      marked[v] = true;
      for (int w : G.adj(v)){
         if (!marked[w])
         {
            dfs(G, w);
            edgeTo[w] = v;
         }
      }
   } 
   
   public boolean hasPathTo(int v)
   {  
      return marked[v];  
   }
   
   public Iterable<Integer> pathTo(int v)
   {
      if (!hasPathTo(v)) return null;
      Stack<Integer> path = new Stack<Integer>();
      for (int x = v; x != s; x = edgeTo[x])
         path.push(x);
      path.push(s);
      return path;
   }

}

Bredth First Search

Algorithm

Repeat until queue is empty

• Remove vertex $v$ from queue.

• Add to queue all unmarked vertices adjacent to $v$ and mark them.

Java Implementation

public class BreadthFirstPaths
{
   private boolean[] marked;
   private int[] edgeTo;
   
   private void bfs(Graph G, int s)
   {
       Queue<Integer> q = new Queue<Integer>();
       q.enqueue(s);
       marked[s] = true;
       while (!q.isEmpty())
       {
          int v = q.dequeue();
          for (int w : G.adj(v))
          {
             if (!marked[w])
             {
                q.enqueue(w);
                marked[w] = true;
                edgeTo[w] = v;
             } 
          }
       }
   }
 }

Connected Components

A connected component is a maximal set of connected vertices.

Connected Components

Algorithm

To visit a vertex v :

• Mark vertex v as visited.

• Recursively visit all unmarked vertices adjacent to v.

Java Implementation

public class CC {
    private boolean[] marked;
    private int[] id;
    private int count;
    
    public CC(Graph G){
        marked = new boolean[G.V()];
        id = new int[G.V()];
        
        for(int i=0;i<G.V();i++){
            if(!marked[i])
            {
                dfs(G, i);
                 count++;
            }
        }
    }
    
    public int count()
    {  
        return count;  
    }
    
    public int id(int v)
    {  
        return id[v];  
    }
    
    // All vertices discovered in same call of dfs have same id
    private void dfs(Graph G, int v)
    {
       marked[v] = true;
       id[v] = count;
       for (int w : G.adj(v))
          if (!marked[w])
              dfs(G, w);
    }
}

Undirected Graph Applications