Tree, Back, Edge and Cross Edges in DFS of Graph

Consider a directed graph given in below, DFS of the below graph is 1 2 4 6 3 5 7 8. In below diagram if DFS is applied on this graph a tree is obtained which is connected using green edges.

• Tree Edge: It is an edge which is present in the tree obtained after applying DFS on the graph. All the Green edges are tree edges. 
• Forward Edge: It is an edge (u, v) such that v is a descendant but not part of the DFS tree. An edge from 1 to 8 is a forward edge. 
• Back edge: It is an edge (u, v) such that v is the ancestor of node u but is not part of the DFS tree. Edge from 6 to 2 is a back edge. Presence of back edge indicates a cycle in directed graph. 
• Cross Edge: It is an edge that connects two nodes such that they do not have any ancestor and a descendant relationship between them. The edge from node 5 to 4 is a cross edge.

Time Complexity(DFS):

Since all the nodes and vertices are visited, the average time complexity for DFS on a graph is O(V + E), where V is the number of vertices and E is the number of edges. In case of DFS on a tree, the time complexity is O(V), where V is the number of nodes.

Algorithm(DFS):

• Pick any node. If it is unvisited, mark it as visited and recur on all its adjacent nodes.
• Repeat until all the nodes are visited, or the node to be searched is found.

Example: Implement DFS using an adjacency list take a directed graph of size n=10, and randomly select the number of edges in the graph varying from 9 to 45. Identify each edge as the forwarding edge, tree edge, back edge, and cross edge.

Python3

# code
import random


class Graph:
    # instance variables
    def __init__(self, v):
        # v is the number of nodes/vertices
        self.time = 0
        self.traversal_array = []
        self.v = v
        # e is the number of edge (randomly chosen between 9 to 45)
        self.e = random.randint(9, 45)
        # adj. list for graph
        self.graph_list = [[] for _ in range(v)]
        # adj. matrix for graph
        self.graph_matrix = [[0 for _ in range(v)] for _ in range(v)]

    # function to create random graph
    def create_random_graph(self):
        # add edges upto e
        for i in range(self.e):
            # choose src and dest of each edge randomly
            src = random.randrange(0, self.v)
            dest = random.randrange(0, self.v)
            # re-choose if src and dest are same or src and dest already has an edge
            while src == dest and self.graph_matrix[src][dest] == 1:
                src = random.randrange(0, self.v)
                dest = random.randrange(0, self.v)
            # add the edge to graph
            self.graph_list[src].append(dest)
            self.graph_matrix[src][dest] = 1

    # function to print adj list
    def print_graph_list(self):
        print("Adjacency List Representation:")
        for i in range(self.v):
            print(i, "-->", *self.graph_list[i])
        print()

    # function to print adj matrix
    def print_graph_matrix(self):
        print("Adjacency Matrix Representation:")
        for i in self.graph_matrix:
            print(i)
        print()

    # function the get number of edges
    def number_of_edges(self):
        return self.e

    # function for dfs
    def dfs(self):
        self.visited = [False]*self.v
        self.start_time = [0]*self.v
        self.end_time = [0]*self.v

        for node in range(self.v):
            if not self.visited[node]:
                self.traverse_dfs(node)
        print()
        print("DFS Traversal: ", self.traversal_array)
        print()

    def traverse_dfs(self, node):
        # mark the node visited
        self.visited[node] = True
        # add the node to traversal
        self.traversal_array.append(node)
        # get the starting time
        self.start_time[node] = self.time
        # increment the time by 1
        self.time += 1
        # traverse through the neighbours
        for neighbour in self.graph_list[node]:
            # if a node is not visited
            if not self.visited[neighbour]:
                # marks the edge as tree edge
                print('Tree Edge:', str(node)+'-->'+str(neighbour))
                # dfs from that node
                self.traverse_dfs(neighbour)
            else:
                # when the parent node is traversed after the neighbour node
                if self.start_time[node] > self.start_time[neighbour] and self.end_time[node] < self.end_time[neighbour]:
                    print('Back Edge:', str(node)+'-->'+str(neighbour))
                # when the neighbour node is a descendant but not a part of tree
                elif self.start_time[node] < self.start_time[neighbour] and self.end_time[node] > self.end_time[neighbour]:
                    print('Forward Edge:', str(node)+'-->'+str(neighbour))
                # when parent and neighbour node do not have any ancestor and a descendant relationship between them
                elif self.start_time[node] > self.start_time[neighbour] and self.end_time[node] > self.end_time[neighbour]:
                    print('Cross Edge:', str(node)+'-->'+str(neighbour))
            self.end_time[node] = self.time
            self.time += 1


if __name__ == "__main__":
    n = 10
    g = Graph(n)
    g.create_random_graph()
    g.print_graph_list()
    g.print_graph_matrix()
    g.dfs()

Output

Adjacency List Representation:
0 --> 5
1 --> 3 7
2 --> 4 3 8 9
3 --> 3
4 --> 0
5 --> 2 0
6 --> 0
7 --> 7 4 3
8 --> 8 9
9 --> 9

Adjacency Matrix Representation:
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
[0, 0, 0, 1, 0, 0, 0, 1, 0, 0]
[0, 0, 0, 1, 1, 0, 0, 0, 1, 1]
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[1, 0, 1, 0, 0, 0, 0, 0, 0, 0]
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 1, 1, 0, 0, 1, 0, 0]
[0, 0, 0, 0, 0, 0, 0, 0, 1, 1]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1]

Tree Edge: 0-->5
Tree Edge: 5-->2
Tree Edge: 2-->4
Tree Edge: 2-->3
Tree Edge: 2-->8
Tree Edge: 8-->9
Forward Edge: 2-->9
Cross Edge: 5-->0
Back Edge: 1-->3
Tree Edge: 1-->7
Cross Edge: 7-->4
Cross Edge: 7-->3
Back Edge: 6-->0

DFS Traversal:  [0, 5, 2, 4, 3, 8, 9, 1, 7, 6]