This is a Python program to find if an undirected graph is bipartite using DFS.

The program creates a graph object and allows the user to determine whether the graph is bipartite.

1. Create classes for Graph and Vertex.

2. Create a function is_bipartite_helper that takes a Vertex object v, a set visited and a dictionary colour as arguments.

3. The function works by painting each vertex a colour opposite to the colour of its neighbours. If it is found not possible to do so, the graph is not bipartite.

4. The function begins by adding v to visited.

5. For each neighbour of v that has not been visited, its colour is set to the colour opposite of v’s colour and is_bipartite_helper on the neighbour is called to see whether it returns False. If so, the graph is not bipartite and False is returned.

6. If a neghbour has already been visited, then it is checked to see whether the colour of the neighbour is opposite to the colour of v. If not, False is returned.

7. After the loop is finished, True is returned to indicate that the graph is bipartite.

8. Create a function is_bipartite that takes a Vertex object v and an empty set visited as arguments.

9. It creates a dictionary colour with v mapped to 0. The values 0 and 1 are used to represent the two different colours.

10. It calls is_bipartite_helper with v, the set visited and the dictionary colour. Its returned value is the return value of this function.

11. Thus, this function returns True if the connected component containing v is bipartite and puts all vertices reachable from v in the set visited.

Here is the source code of a Python program to find if an undirected graph is bipartite using DFS. The program output is shown below.

class Graph: def __init__(self): # dictionary containing keys that map to the corresponding vertex object self.vertices = {} def add_vertex(self, key): """Add a vertex with the given key to the graph.""" vertex = Vertex(key) self.vertices[key] = vertex def get_vertex(self, key): """Return vertex object with the corresponding key.""" return self.vertices[key] def __contains__(self, key): return key in self.vertices def add_edge(self, src_key, dest_key, weight=1): """Add edge from src_key to dest_key with given weight.""" self.vertices[src_key].add_neighbour(self.vertices[dest_key], weight) def add_undirected_edge(self, v1_key, v2_key, weight=1): """Add undirected edge (2 directed edges) between v1_key and v2_key with given weight.""" self.add_edge(v1_key, v2_key, weight) self.add_edge(v2_key, v1_key, weight) def does_undirected_edge_exist(self, v1_key, v2_key): """Return True if there is an undirected edge between v1_key and v2_key.""" return (self.does_edge_exist(v1_key, v2_key) and self.does_edge_exist(v1_key, v2_key)) def does_edge_exist(self, src_key, dest_key): """Return True if there is an edge from src_key to dest_key.""" return self.vertices[src_key].does_it_point_to(self.vertices[dest_key]) def __iter__(self): return iter(self.vertices.values()) class Vertex: def __init__(self, key): self.key = key self.points_to = {} def get_key(self): """Return key corresponding to this vertex object.""" return self.key def add_neighbour(self, dest, weight): """Make this vertex point to dest with given edge weight.""" self.points_to[dest] = weight def get_neighbours(self): """Return all vertices pointed to by this vertex.""" return self.points_to.keys() def get_weight(self, dest): """Get weight of edge from this vertex to dest.""" return self.points_to[dest] def does_it_point_to(self, dest): """Return True if this vertex points to dest.""" return dest in self.points_to def is_bipartite(vertex, visited): """Return True if component containing vertex is bipartite and put all vertices in its component in set visited.""" colour = {vertex: 0} return is_bipartite_helper(vertex, visited, colour) def is_bipartite_helper(v, visited, colour): """Return True if component containing vertex is bipartite and put all vertices in its component in set visited. Uses dictionary colour to keep track of colour of each vertex.""" visited.add(v) next_colour = 1 - colour[v] # switch colour for dest in v.get_neighbours(): if dest not in visited: colour[dest] = next_colour if not is_bipartite_helper(dest, visited, colour): return False else: if colour[dest] != next_colour: return False return True g = Graph() print('Undirected Graph') print('Menu') print('add vertex <key>') print('add edge <vertex1> <vertex2>') print('bipartite') print('display') print('quit') while True: do = input('What would you like to do? ').split() operation = do[0] if operation == 'add': suboperation = do[1] if suboperation == 'vertex': key = int(do[2]) if key not in g: g.add_vertex(key) else: print('Vertex already exists.') elif suboperation == 'edge': v1 = int(do[2]) v2 = int(do[3]) if v1 not in g: print('Vertex {} does not exist.'.format(v1)) elif v2 not in g: print('Vertex {} does not exist.'.format(v2)) else: if not g.does_undirected_edge_exist(v1, v2): g.add_undirected_edge(v1, v2) else: print('Edge already exists.') elif operation == 'bipartite': bipartite = True visited = set() for v in g: if v not in visited: if not is_bipartite(v, visited): bipartite = False break if bipartite: print('Graph is bipartite.') else: print('Graph is not bipartite.') elif operation == 'display': print('Vertices: ', end='') for v in g: print(v.get_key(), end=' ') print() print('Edges: ') for v in g: for dest in v.get_neighbours(): w = v.get_weight(dest) print('(src={}, dest={}, weight={}) '.format(v.get_key(), dest.get_key(), w)) print() elif operation == 'quit': break

1. An instance of Graph is created.

2. A menu is presented to the user to perform various operations on the graph.

3. To determine whether the graph is bipartite, is_bipartite is called with a vertex from the graph and an empty set visited.

4. If is_bipartite returns True, the graph is not bipartite. Otherwise, if not all vertices were visited, is_bipartite is called again with an unvisited source vertex.

5. This continues until all vertices have been visited or is_bipartite returns False.

6. If all vertices have been visited, the graph is bipartite.

Case 1: Undirected Graph Menu add vertex <key> add edge <vertex1> <vertex2> bipartite display quit What would you like to do? add vertex 1 What would you like to do? add vertex 2 What would you like to do? add vertex 3 What would you like to do? add vertex 4 What would you like to do? add edge 1 2 What would you like to do? bipartite Graph is bipartite. What would you like to do? add edge 3 2 What would you like to do? bipartite Graph is bipartite. What would you like to do? add edge 1 4 What would you like to do? bipartite Graph is bipartite. What would you like to do? add edge 2 4 What would you like to do? bipartite Graph is not bipartite. What would you like to do? quit Case 2: Undirected Graph Menu add vertex <key> add edge <vertex1> <vertex2> bipartite display quit What would you like to do? add vertex 1 What would you like to do? add vertex 2 What would you like to do? add vertex 3 What would you like to do? add vertex 4 What would you like to do? add vertex 5 What would you like to do? add vertex 6 What would you like to do? add edge 1 2 What would you like to do? add edge 2 3 What would you like to do? add edge 3 4 What would you like to do? bipartite Graph is bipartite. What would you like to do? add edge 5 1 What would you like to do? add edge 6 4 What would you like to do? bipartite Graph is bipartite. What would you like to do? add edge 6 5 What would you like to do? bipartite Graph is bipartite. What would you like to do? add edge 6 2 What would you like to do? bipartite Graph is bipartite. What would you like to do? add edge 6 1 What would you like to do? bipartite Graph is not bipartite. What would you like to do? quit

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