# Euler Circuit in a Directed Graph

Eulerian Path is a path in graph that visits every edge exactly once. Eulerian Circuit is an Eulerian Path which starts and ends on the same vertex.

A graph is said to be eulerian if it has eulerian cycle. We have discussed eulerian circuit for an undirected graph. In this post, same is discussed for a directed graph.

For example, the following graph has eulerian cycle as {1, 0, 3, 4, 0, 2, 1} How to check if a directed graph is eulerian?
A directed graph has an eulerian cycle if following conditions are true (Source: Wiki)
1) All vertices with nonzero degree belong to a single strongly connected component.
2) In degree and out degree of every vertex is same.

We can detect singly connected component using Kosaraju’s DFS based simple algorithm.
To compare in degree and out degree, we need to store in degree an out degree of every vertex. Out degree can be obtained by size of adjacency list. In degree can be stored by creating an array of size equal to number of vertices.

Following are C++ and Java implementations of above approach.

## C++

 `// A C++ program to check if a given directed graph is Eulerian or not ` `#include ` `#include ` `#define CHARS 26 ` `using` `namespace` `std; ` ` `  `// A class that represents an undirected graph ` `class` `Graph ` `{ ` `    ``int` `V;    ``// No. of vertices ` `    ``list<``int``> *adj;    ``// A dynamic array of adjacency lists ` `    ``int` `*in; ` `public``: ` `    ``// Constructor and destructor ` `    ``Graph(``int` `V); ` `    ``~Graph()   { ``delete` `[] adj; ``delete` `[] in; } ` ` `  `    ``// function to add an edge to graph ` `    ``void` `addEdge(``int` `v, ``int` `w) { adj[v].push_back(w);  (in[w])++; } ` ` `  `    ``// Method to check if this graph is Eulerian or not ` `    ``bool` `isEulerianCycle(); ` ` `  `    ``// Method to check if all non-zero degree vertices are connected ` `    ``bool` `isSC(); ` ` `  `    ``// Function to do DFS starting from v. Used in isConnected(); ` `    ``void` `DFSUtil(``int` `v, ``bool` `visited[]); ` ` `  `    ``Graph getTranspose(); ` `}; ` ` `  `Graph::Graph(``int` `V) ` `{ ` `    ``this``->V = V; ` `    ``adj = ``new` `list<``int``>[V]; ` `    ``in = ``new` `int``[V]; ` `    ``for` `(``int` `i = 0; i < V; i++) ` `       ``in[i] = 0; ` `} ` ` `  `/* This function returns true if the directed graph has an eulerian ` `   ``cycle, otherwise returns false  */` `bool` `Graph::isEulerianCycle() ` `{ ` `    ``// Check if all non-zero degree vertices are connected ` `    ``if` `(isSC() == ``false``) ` `        ``return` `false``; ` ` `  `    ``// Check if in degree and out degree of every vertex is same ` `    ``for` `(``int` `i = 0; i < V; i++) ` `        ``if` `(adj[i].size() != in[i]) ` `            ``return` `false``; ` ` `  `    ``return` `true``; ` `} ` ` `  `// A recursive function to do DFS starting from v ` `void` `Graph::DFSUtil(``int` `v, ``bool` `visited[]) ` `{ ` `    ``// Mark the current node as visited and print it ` `    ``visited[v] = ``true``; ` ` `  `    ``// Recur for all the vertices adjacent to this vertex ` `    ``list<``int``>::iterator i; ` `    ``for` `(i = adj[v].begin(); i != adj[v].end(); ++i) ` `        ``if` `(!visited[*i]) ` `            ``DFSUtil(*i, visited); ` `} ` ` `  `// Function that returns reverse (or transpose) of this graph ` `// This function is needed in isSC() ` `Graph Graph::getTranspose() ` `{ ` `    ``Graph g(V); ` `    ``for` `(``int` `v = 0; v < V; v++) ` `    ``{ ` `        ``// Recur for all the vertices adjacent to this vertex ` `        ``list<``int``>::iterator i; ` `        ``for``(i = adj[v].begin(); i != adj[v].end(); ++i) ` `        ``{ ` `            ``g.adj[*i].push_back(v); ` `            ``(g.in[v])++; ` `        ``} ` `    ``} ` `    ``return` `g; ` `} ` ` `  `// This function returns true if all non-zero degree vertices of  ` `// graph are strongly connected (Please refer  ` `// https://tutorialspoint.dev/slugresolver/connectivity-in-a-directed-graph/ ) ` `bool` `Graph::isSC() ` `{ ` `    ``// Mark all the vertices as not visited (For first DFS) ` `    ``bool` `visited[V]; ` `    ``for` `(``int` `i = 0; i < V; i++) ` `        ``visited[i] = ``false``; ` ` `  `    ``// Find the first vertex with non-zero degree ` `    ``int` `n; ` `    ``for` `(n = 0; n < V; n++) ` `        ``if` `(adj[n].size() > 0) ` `          ``break``; ` ` `  `    ``// Do DFS traversal starting from first non zero degree vertex. ` `    ``DFSUtil(n, visited); ` ` `  `     ``// If DFS traversal doesn't visit all vertices, then return false. ` `    ``for` `(``int` `i = 0; i < V; i++) ` `        ``if` `(adj[i].size() > 0 && visited[i] == ``false``) ` `              ``return` `false``; ` ` `  `    ``// Create a reversed graph ` `    ``Graph gr = getTranspose(); ` ` `  `    ``// Mark all the vertices as not visited (For second DFS) ` `    ``for` `(``int` `i = 0; i < V; i++) ` `        ``visited[i] = ``false``; ` ` `  `    ``// Do DFS for reversed graph starting from first vertex. ` `    ``// Staring Vertex must be same starting point of first DFS ` `    ``gr.DFSUtil(n, visited); ` ` `  `    ``// If all vertices are not visited in second DFS, then ` `    ``// return false ` `    ``for` `(``int` `i = 0; i < V; i++) ` `        ``if` `(adj[i].size() > 0 && visited[i] == ``false``) ` `             ``return` `false``; ` ` `  `    ``return` `true``; ` `} ` ` `  `// Driver program to test above functions ` `int` `main() ` `{ ` `    ``// Create a graph given in the above diagram ` `    ``Graph g(5); ` `    ``g.addEdge(1, 0); ` `    ``g.addEdge(0, 2); ` `    ``g.addEdge(2, 1); ` `    ``g.addEdge(0, 3); ` `    ``g.addEdge(3, 4); ` `    ``g.addEdge(4, 0); ` ` `  `    ``if` `(g.isEulerianCycle()) ` `       ``cout << ``"Given directed graph is eulerian n"``; ` `    ``else` `       ``cout << ``"Given directed graph is NOT eulerian n"``; ` `    ``return` `0; ` `} `

## Java

 `// A Java program to check if a given directed graph is Eulerian or not ` ` `  `// A class that represents an undirected graph ` `import` `java.io.*; ` `import` `java.util.*; ` `import` `java.util.LinkedList; ` ` `  `// This class represents a directed graph using adjacency list ` `class` `Graph ` `{ ` `    ``private` `int` `V;   ``// No. of vertices ` `    ``private` `LinkedList adj[];``//Adjacency List ` `    ``private` `int` `in[];            ``//maintaining in degree ` ` `  `    ``//Constructor ` `    ``Graph(``int` `v) ` `    ``{ ` `        ``V = v; ` `        ``adj = ``new` `LinkedList[v]; ` `        ``in = ``new` `int``[V]; ` `        ``for` `(``int` `i=``0``; i i =adj[v].iterator(); ` `        ``while` `(i.hasNext()) ` `        ``{ ` `            ``n = i.next(); ` `            ``if` `(!visited[n]) ` `                ``DFSUtil(n,visited); ` `        ``} ` `    ``} ` ` `  `    ``// Function that returns reverse (or transpose) of this graph ` `    ``Graph getTranspose() ` `    ``{ ` `        ``Graph g = ``new` `Graph(V); ` `        ``for` `(``int` `v = ``0``; v < V; v++) ` `        ``{ ` `            ``// Recur for all the vertices adjacent to this vertex ` `            ``Iterator i = adj[v].listIterator(); ` `            ``while` `(i.hasNext()) ` `            ``{ ` `                ``g.adj[i.next()].add(v); ` `                ``(g.in[v])++; ` `            ``} ` `        ``} ` `        ``return` `g; ` `    ``} ` ` `  `    ``// The main function that returns true if graph is strongly ` `    ``// connected ` `    ``Boolean isSC() ` `    ``{ ` `        ``// Step 1: Mark all the vertices as not visited (For ` `        ``// first DFS) ` `        ``Boolean visited[] = ``new` `Boolean[V]; ` `        ``for` `(``int` `i = ``0``; i < V; i++) ` `            ``visited[i] = ``false``; ` ` `  `        ``// Step 2: Do DFS traversal starting from first vertex. ` `        ``DFSUtil(``0``, visited); ` ` `  `        ``// If DFS traversal doesn't visit all vertices, then return false. ` `        ``for` `(``int` `i = ``0``; i < V; i++) ` `            ``if` `(visited[i] == ``false``) ` `                ``return` `false``; ` ` `  `        ``// Step 3: Create a reversed graph ` `        ``Graph gr = getTranspose(); ` ` `  `        ``// Step 4: Mark all the vertices as not visited (For second DFS) ` `        ``for` `(``int` `i = ``0``; i < V; i++) ` `            ``visited[i] = ``false``; ` ` `  `        ``// Step 5: Do DFS for reversed graph starting from first vertex. ` `        ``// Staring Vertex must be same starting point of first DFS ` `        ``gr.DFSUtil(``0``, visited); ` ` `  `        ``// If all vertices are not visited in second DFS, then ` `        ``// return false ` `        ``for` `(``int` `i = ``0``; i < V; i++) ` `            ``if` `(visited[i] == ``false``) ` `                ``return` `false``; ` ` `  `        ``return` `true``; ` `    ``} ` ` `  `    ``/* This function returns true if the directed graph has an eulerian ` `       ``cycle, otherwise returns false  */` `    ``Boolean isEulerianCycle() ` `    ``{ ` `        ``// Check if all non-zero degree vertices are connected ` `        ``if` `(isSC() == ``false``) ` `            ``return` `false``; ` ` `  `        ``// Check if in degree and out degree of every vertex is same ` `        ``for` `(``int` `i = ``0``; i < V; i++) ` `            ``if` `(adj[i].size() != in[i]) ` `                ``return` `false``; ` ` `  `        ``return` `true``; ` `    ``} ` ` `  `    ``public` `static` `void` `main (String[] args) ``throws` `java.lang.Exception ` `    ``{ ` `        ``Graph g = ``new` `Graph(``5``); ` `        ``g.addEdge(``1``, ``0``); ` `        ``g.addEdge(``0``, ``2``); ` `        ``g.addEdge(``2``, ``1``); ` `        ``g.addEdge(``0``, ``3``); ` `        ``g.addEdge(``3``, ``4``); ` `        ``g.addEdge(``4``, ``0``); ` ` `  `        ``if` `(g.isEulerianCycle()) ` `            ``System.out.println(``"Given directed graph is eulerian "``); ` `        ``else` `            ``System.out.println(``"Given directed graph is NOT eulerian "``); ` `    ``} ` `} ` `//This code is contributed by Aakash Hasija `

## Python

 `# A Python program to check if a given  ` `# directed graph is Eulerian or not ` ` `  `from` `collections ``import` `defaultdict ` ` `  `class` `Graph(): ` ` `  `    ``def` `__init__(``self``, vertices): ` `        ``self``.V ``=` `vertices ` `        ``self``.graph ``=` `defaultdict(``list``) ` `        ``self``.IN ``=` `[``0``] ``*` `vertices ` ` `  `    ``def` `addEdge(``self``, v, u): ` ` `  `        ``self``.graph[v].append(u) ` `        ``self``.IN[u] ``+``=` `1` ` `  `    ``def` `DFSUtil(``self``, v, visited): ` `        ``visited[v] ``=` `True` `        ``for` `node ``in` `self``.graph[v]: ` `            ``if` `visited[node] ``=``=` `False``: ` `                ``self``.DFSUtil(node, visited) ` ` `  `    ``def` `getTranspose(``self``): ` `        ``gr ``=` `Graph(``self``.V) ` ` `  `        ``for` `node ``in` `range``(``self``.V): ` `            ``for` `child ``in` `self``.graph[node]: ` `                ``gr.addEdge(child, node) ` ` `  `        ``return` `gr ` ` `  `    ``def` `isSC(``self``): ` `        ``visited ``=` `[``False``] ``*` `self``.V ` ` `  `        ``v ``=` `0` `        ``for` `v ``in` `range``(``self``.V): ` `            ``if` `len``(``self``.graph[v]) > ``0``: ` `                ``break` ` `  `        ``self``.DFSUtil(v, visited) ` ` `  `        ``# If DFS traversal doesn't visit all  ` `        ``# vertices, then return false. ` `        ``for` `i ``in` `range``(``self``.V): ` `            ``if` `visited[i] ``=``=` `False``: ` `                ``return` `False` ` `  `        ``gr ``=` `self``.getTranspose() ` ` `  `        ``visited ``=` `[``False``] ``*` `self``.V ` `        ``gr.DFSUtil(v, visited) ` ` `  `        ``for` `i ``in` `range``(``self``.V): ` `            ``if` `visited[i] ``=``=` `False``: ` `                ``return` `False` ` `  `        ``return` `True` ` `  `    ``def` `isEulerianCycle(``self``): ` ` `  `        ``# Check if all non-zero degree vertices  ` `        ``# are connected ` `        ``if` `self``.isSC() ``=``=` `False``: ` `            ``return` `False` ` `  `        ``# Check if in degree and out degree of  ` `        ``# every vertex is same ` `        ``for` `v ``in` `range``(``self``.V): ` `            ``if` `len``(``self``.graph[v]) !``=` `self``.IN[v]: ` `                ``return` `False` ` `  `        ``return` `True` ` `  ` `  `g ``=` `Graph(``5``); ` `g.addEdge(``1``, ``0``); ` `g.addEdge(``0``, ``2``); ` `g.addEdge(``2``, ``1``); ` `g.addEdge(``0``, ``3``); ` `g.addEdge(``3``, ``4``); ` `g.addEdge(``4``, ``0``); ` `if` `g.isEulerianCycle(): ` `   ``print` `"Given directed graph is eulerian"``; ` `else``: ` `   ``print` `"Given directed graph is NOT eulerian"``; ` ` `  `# This code is contributed by Divyanshu Mehta `

Output:

`Given directed graph is eulerian `

Time complexity of the above implementation is O(V + E) as Kosaraju’s algorithm takes O(V + E) time. After running Kosaraju’s algorithm we traverse all vertices and compare in degree with out degree which takes O(V) time.

See following as an application of this.
Find if the given array of strings can be chained to form a circle.