Convert the undirected graph into directed graph such that there is no path of length greater than 1

Given an undirected graph with N vertices and M edges and no self loops or multiple edges. The task is to convert the given undirected graph into a directed graph such that there is no path of length greater than 1. If it is possible to make such a graph then print two space-separated integers u and v in M lines where u, v denotes source and destination vertices respectively. If not possible then print -1. Examples:

Input: Output: 1 2 1 3 1 4 Input: Output: -1 For the given graph it is not possible to get a directed graph such that there is no path of length greater than 1

Approach: Let suppose the graph contains a cycle of odd length. It means that some two consecutive edges of this cycle will be oriented in the same way and will form a path of length two. Then the answer is -1. And if the graph contains no cycles of odd length. Then it is bipartite. Let’s color it and see what we got. We got some vertices in the left part, some vertices in the right part and all edges connecting vertices from different parts. Let’s orient all edges such that they will go from the left part to the right part. Below is the implementation of the above approach: 

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
#define N 100005

// To store the graph
vector<int> gr[N];

// To store colour of each vertex
int colour[N];

// To store edges
vector<pair<int, int> > edges;

// To check graph is bipartite or not
bool bip;

// Function to add edges
void add_edge(int x, int y)
{
    gr[x].push_back(y);
    gr[y].push_back(x);
    edges.push_back(make_pair(x, y));
}

// Function to check given graph
// is bipartite or not
void dfs(int x, int col)
{
    // colour the vertex x
    colour[x] = col;

    // For all it's child vertices
    for (auto i : gr[x]) {
        // If still not visited
        if (colour[i] == -1)
            dfs(i, col ^ 1);

        // If visited and having
        // same colour as parent
        else if (colour[i] == col)
            bip = false;
    }
}

// Function to convert the undirected
// graph into the directed graph such that
// there is no path of length greater than 1
void Directed_Graph(int n, int m)
{

    // Initially each vertex has no colour
    memset(colour, -1, sizeof colour);

    // Suppose bipartite is possible
    bip = true;

    // Call bipartite function
    dfs(1, 1);

    // If bipartite is not possible
    if (!bip) {
        cout << -1;
        return;
    }

    // If bipartite is possible
    for (int i = 0; i < m; i++) {

        // Make an edge from vertex having
        // colour 1 to colour 0
        if (colour[edges[i].first] == 0)
            swap(edges[i].first, edges[i].second);

        cout << edges[i].first << " "
             << edges[i].second << endl;
    }
}

// Driver code
int main()
{
    int n = 4, m = 3;

    // Add edges
    add_edge(1, 2);
    add_edge(1, 3);
    add_edge(1, 4);

    // Function call
    Directed_Graph(n, m);

    return 0;
}

Java

// Java implementation of the approach
import java.util.*;

class GFG
{
static class pair
{
    int first, second;
    public pair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}

static int N = 100005;

// To store the graph
static Vector<Integer> []gr = new Vector[N];

// To store colour of each vertex
static int []colour = new int[N];

// To store edges
static Vector<pair> edges = new Vector<>();

// To check graph is bipartite or not
static boolean bip;

// Function to add edges
static void add_edge(int x, int y)
{
    gr[x].add(y);
    gr[y].add(x);
    edges.add(new pair(x, y));
}

// Function to check given graph
// is bipartite or not
static void dfs(int x, int col)
{
    // colour the vertex x
    colour[x] = col;

    // For all it's child vertices
    for (Integer i : gr[x])
    {
        // If still not visited
        if (colour[i] == -1)
            dfs(i, col ^ 1);

        // If visited and having
        // same colour as parent
        else if (colour[i] == col)
            bip = false;
    }
}

// Function to convert the undirected
// graph into the directed graph such that
// there is no path of length greater than 1
static void Directed_Graph(int n, int m)
{

    // Initially each vertex has no colour
    for (int i = 0; i < N; i++)
        colour[i] = -1;

    // Suppose bipartite is possible
    bip = true;

    // Call bipartite function
    dfs(1, 1);

    // If bipartite is not possible
    if (!bip)
    {
        System.out.print(-1);
        return;
    }

    // If bipartite is possible
    for (int i = 0; i < m; i++)
    {

        // Make an edge from vertex having
        // colour 1 to colour 0
        if (colour[edges.get(i).first] == 0)
        {
            Collections.swap(edges, edges.get(i).first,
                                    edges.get(i).second);
        }

        System.out.println(edges.get(i).first + " " +
                           edges.get(i).second);
    }
}

// Driver code
public static void main(String[] args)
{
    int n = 4, m = 3;
    for (int i = 0; i < N; i++)
        gr[i] = new Vector<>();

    // Add edges
    add_edge(1, 2);
    add_edge(1, 3);
    add_edge(1, 4);

    // Function call
    Directed_Graph(n, m);
}
}

// This code is contributed by PrinciRaj1992

Python3

# Python3 implementation of the approach

N = 100005

# To store the graph
gr = [[] for i in range(N)]

# To store colour of each vertex
colour = [-1] * N

# To store edges
edges = []

# To check graph is bipartite or not
bip = True

# Function to add edges
def add_edge(x, y):

    gr[x].append(y)
    gr[y].append(x)
    edges.append((x, y))

# Function to check given graph
# is bipartite or not
def dfs(x, col):

    # colour the vertex x
    colour[x] = col
    global bip

    # For all it's child vertices
    for i in gr[x]:
        # If still not visited
        if colour[i] == -1:
            dfs(i, col ^ 1)

        # If visited and having
        # same colour as parent
        elif colour[i] == col:
            bip = False

# Function to convert the undirected
# graph into the directed graph such that
# there is no path of length greater than 1
def Directed_Graph(n, m):

    # Call bipartite function
    dfs(1, 1)

    # If bipartite is not possible
    if not bip:
        print(-1)
        return

    # If bipartite is possible
    for i in range(0, m):

        # Make an edge from vertex
        # having colour 1 to colour 0
        if colour[edges[i][0]] == 0:
            edges[i][0], edges[i][1] = edges[i][1], edges[i][0]

        print(edges[i][0], edges[i][1])

# Driver code
if __name__ == "__main__":

    n, m = 4, 3

    # Add edges
    add_edge(1, 2)
    add_edge(1, 3)
    add_edge(1, 4)

    # Function call
    Directed_Graph(n, m)

# This code is contributed by Rituraj Jain

C#

// C# implementation of the approach
using System;
using System.Collections.Generic;

class GFG
{

class pair
{
    public int first, second;
    public pair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}

static int N = 100005;

// To store the graph
static List<int> []gr = new List<int>[N];

// To store colour of each vertex
static int []colour = new int[N];

// To store edges
static List<pair> edges = new List<pair>();

// To check graph is bipartite or not
static Boolean bip;

// Function to add edges
static void add_edge(int x, int y)
{
    gr[x].Add(y);
    gr[y].Add(x);
    edges.Add(new pair(x, y));
}

// Function to check given graph
// is bipartite or not
static void dfs(int x, int col)
{
    // colour the vertex x
    colour[x] = col;

    // For all it's child vertices
    foreach (int i in gr[x])
    {
        // If still not visited
        if (colour[i] == -1)
            dfs(i, col ^ 1);

        // If visited and having
        // same colour as parent
        else if (colour[i] == col)
            bip = false;
    }
}

// Function to convert the undirected
// graph into the directed graph such that
// there is no path of length greater than 1
static void Directed_Graph(int n, int m)
{

    // Initially each vertex has no colour
    for (int i = 0; i < N; i++)
        colour[i] = -1;

    // Suppose bipartite is possible
    bip = true;

    // Call bipartite function
    dfs(1, 1);

    // If bipartite is not possible
    if (!bip)
    {
        Console.Write(-1);
        return;
    }

    // If bipartite is possible
    for (int i = 0; i < m; i++)
    {

        // Make an edge from vertex having
        // colour 1 to colour 0
        if (colour[edges[i].first] == 0)
        {
            var v = edges[i].first;
            edges[i].first = edges[i].second;
            edges[i].second = v;
        }

        Console.WriteLine(edges[i].first + " " +
                          edges[i].second);
    }
}

// Driver code
public static void Main(String[] args)
{
    int n = 4, m = 3;
    for (int i = 0; i < N; i++)
        gr[i] = new List<int>();

    // Add edges
    add_edge(1, 2);
    add_edge(1, 3);
    add_edge(1, 4);

    // Function call
    Directed_Graph(n, m);
}
}

// This code is contributed by Rajput-Ji

Javascript

<script>

// JavaScript code to implement the above approach
// Python3 implementation of the approach
let N = 100005

// To store the graph
let gr = new Array(N);
for(let i = 0; i < N; i++){
    gr[i] = new Array()
}

// To store colour of each vertex
let colour = new Array(N).fill(-1)

// To store edges
let edges = []

// To check graph is bipartite or not
let bip = true

// Function to add edges
function add_edge(x, y){

    gr[x].push(y)
    gr[y].push(x)
    edges.push([x, y])
}

// Function to check given graph
// is bipartite or not
function dfs(x, col){

    // colour the vertex x
    colour[x] = col

    // For all it's child vertices
    for(let i of gr[x]){
        // If still not visited
        if(colour[i] == -1)
            dfs(i, col ^ 1)

        // If visited and having
        // same colour as parent
        else if(colour[i] == col)
            bip = false
    }
}

// Function to convert the undirected
// graph into the directed graph such that
// there is no path of length greater than 1
function Directed_Graph(n, m){

    // Call bipartite function
    dfs(1, 1)

    // If bipartite is not possible
    if(bip == 0){
        document.write(-1,"</br>")
        return
    }

    // If bipartite is possible
    for(let i=0;i<m;i++){

        // Make an edge from vertex
        // having colour 1 to colour 0
        if(colour[edges[i][0]] == 0){
            let temp = edges[i][0]
            edges[i][0] = edges[i][1]
            edges[i][1] = temp
        }

        document.write(edges[i][0], edges[i][1],"</br>")
    }
}
// Driver code

let n =4, m = 3

// Add edges
add_edge(1, 2)
add_edge(1, 3)
add_edge(1, 4)

// Function call
Directed_Graph(n, m)

// This code is contributed by shinjanpatra

</script>

Output:

1 2
1 3
1 4

Time Complexity: O(V + E)

The time complexity of the above approach is O(V + E) where V is the number of vertices and E is the number of edges in the given graph. Since we are traversing the graph using DFS, the time complexity is proportional to the number of vertices plus the number of edges in the graph.

Space Complexity: O(V + E)

The space complexity of the above approach is also O(V + E) as we are using an adjacency list to store the graph and also a vector to store the edges.