class GraphTests(unittest.TestCase):
def setUp(self):
self.g = Graph()
for v in range(7):
self.g.addVertex(v)
self.g.addEdge(0, 1, 6)
self.g.addEdge(0, 2, 4)
self.g.addEdge(1, 2, 3)
self.g.addEdge(1, 5, 7)
self.g.addEdge(2, 3, 9)
self.g.addEdge(2, 4, 1)
self.g.addEdge(3, 4, 1)
self.g.addEdge(4, 5, 5)
self.g.addEdge(4, 6, 2)
def tearDown(self):
self.g = None
def testBasic(self):
for v in self.g.getVertices():
self.assertTrue(v in range(7))
def testNeighbours(self):
zero_s_nbrs = [1, 2]
for v in zero_s_nbrs:
self.assertTrue(v in self.g.getNeighbours(0))
one_s_nbrs = [2, 5]
for v in one_s_nbrs:
self.assertTrue(v in self.g.getNeighbours(1))
two_s_nbrs = [3, 4]
for v in two_s_nbrs:
self.assertTrue(v in self.g.getNeighbours(2))
three_s_nbrs = [4]
for v in three_s_nbrs:
self.assertTrue(v in self.g.getNeighbours(3))
four_s_nbrs = [5, 6]
for v in four_s_nbrs:
self.assertTrue(v in self.g.getNeighbours(4))
def testEdges(self):
expected_edges = [
(0, 1, 6),
(0, 2, 4),
(1, 2, 3),
(1, 5, 7),
(2, 3, 9),
(2, 4, 1),
(3, 4, 1),
(4, 5, 5),
(4, 6, 2)
]
edges = self.g.getEdges()
for edge in expected_edges:
self.assertTrue(edge in edges)
def testIsolatedVertex(self):
self.g.addVertex(10)
isolated = self.g.findIsolated()
self.assertTrue(10 in isolated)
def testGetPath(self):
path = self.g.getPath(0, 6, [])
self.assertEquals(path, [0, 1, 2, 3, 4, 6])
path = self.g.getPath(0, 10, [])
self.assertEquals(None, path)
def testGetAllPaths(self):
expectedPaths = [[0, 1, 2, 3, 4, 6],
[0, 1, 2, 4, 6],
[0, 2, 3, 4, 6],
[0, 2, 4, 6]
]
paths = self.g.getAllPaths(0, 6, [])
for path in expectedPaths:
self.assertTrue(path in paths)
def testInDegree(self):
self.g.addVertex(10)
expectedDegrees = [(0, 0), # 0 vertex has 0 edges going to it
(1, 1),
(2, 2),
(3, 1),
(4, 2),
(5, 2),
(6, 1),
(10, 0)
]
for (vertex, inDegree) in expectedDegrees:
self.assertEqual(self.g.inDegree(vertex), inDegree)
def testOutDegree(self):
self.g.addVertex(10)
expectedDegrees = [(0, 2), # 0 vertex has 2 edges going from it
(1, 2),
(2, 2),
(3, 1),
(4, 2),
(5, 0),
(6, 0),
(10, 0)
]
for (vertex, outDegree) in expectedDegrees:
self.assertEqual(self.g.outDegree(vertex), outDegree)
def testDegree(self):
self.g.addVertex(10)
expectedDegrees = [(0, 2), # 0 vertex has 2 edges
(1, 3),
(2, 4),
(3, 2),
(4, 4),
(5, 2),
(6, 1),
(10, 0)
]
for (vertex, degree) in expectedDegrees:
self.assertEqual(self.g.getDegree(vertex), degree)
def testDegreeSumFormula(self):
self.assertTrue(self.g.verifyDegreeSumFormula())
def testMinDegreeOfGraphV(self):
self.assertEqual(self.g.delta(), 1)
#add a isolated vertex and check the degree 0
self.g.addVertex(10)
self.assertEqual(self.g.delta(), 0)
def testMaxDegreeOfGraphV(self):
self.assertEqual(self.g.Delta(), 4)
def testDegreeSequence(self):
expected = (4, 4, 3, 2, 2, 2, 1)
self.assertEqual(self.g.degreeSequence(), expected)
def testErdosGallaiTheorem(self):
self.assertTrue(Graph.isGraphicSequence([2, 2, 2, 2, 1, 1]))
self.assertTrue(Graph.isGraphicSequence([3, 3, 3, 3, 3, 3]))
self.assertTrue(Graph.isGraphicSequence([3, 3, 2, 1, 1]))
self.assertFalse(Graph.isGraphicSequence([4, 3, 2, 2, 2, 1, 1]))
self.assertFalse(Graph.isGraphicSequence([6, 6, 5, 4, 4, 2, 1]))
self.assertFalse(Graph.isGraphicSequence([3, 3, 3, 1]))
def testHavelHakimiAlgorithm(self):
self.assertTrue(Graph.isGraphicSequenceIterative([2, 2, 2, 2, 1, 1]))
self.assertTrue(Graph.isGraphicSequenceIterative([3, 3, 3, 3, 3, 3]))
self.assertTrue(Graph.isGraphicSequenceIterative([3, 3, 2, 1, 1]))
self.assertFalse(Graph.isGraphicSequenceIterative([4, 3, 2, 2, 2, 1, 1]))
self.assertFalse(Graph.isGraphicSequenceIterative([6, 6, 5, 4, 4, 2, 1]))
self.assertFalse(Graph.isGraphicSequenceIterative([3, 3, 3, 1]))
def testDensity(self):
g = Graph()
vertices = ['a', 'b', 'c', 'd', 'e', 'f']
for v in vertices:
g.addVertex(v)
"""
g = { "a" : ["d","f"],
"b" : ["c","b"],
"c" : ["b", "c", "d", "e"],
"d" : ["a", "c"],
"e" : ["c"],
"f" : ["a"]
}
"""
g.addEdge('a', 'd')
g.addEdge('a', 'f')
g.addEdge('b', 'c')
g.addEdge('b', 'b')
g.addEdge('c', 'c')
g.addEdge('c', 'd')
g.addEdge('c', 'e')
self.assertAlmostEqual(0.466666666667, g.density(), places=7)
"""
complete_graph = {
"a" : ["b","c"],
"b" : ["a","c"],
"c" : ["a","b"]
}
"""
complete_graph = Graph()
vertices = ['a', 'b', 'c']
for v in vertices:
complete_graph.addVertex(v)
complete_graph.addEdge('a', 'b')
complete_graph.addEdge('a', 'c')
complete_graph.addEdge('b', 'c')
self.assertEqual(1.0, complete_graph.density())
"""
isolated_graph = {
"a" : [],
"b" : [],
"c" : []
}
"""
isolated_graph = Graph()
vertices = ['a', 'b', 'c']
for v in vertices:
isolated_graph.addVertex(v)
self.assertEqual(0.0, isolated_graph.density())
def testIsConnected(self):
self.assertTrue(self.g.isConnected())
self.g.addVertex(10)
self.assertFalse(self.g.isConnected())
"""
g = { "a" : ["d"],
"b" : ["c"],
"c" : ["b", "c", "d", "e"],
"d" : ["a", "c"],
"e" : ["c"],
"f" : []
}
"""
g = Graph()
vertices = ['a', 'b', 'c', 'd', 'e', 'f']
for v in vertices:
g.addVertex(v)
g.addEdge('a', 'd')
g.addEdge('b', 'c')
g.addEdge('c', 'd')
g.addEdge('c', 'e')
self.assertFalse(g.isConnected('a'))
"""
g2 = {
"a" : ["d","f"],
"b" : ["c"],
"c" : ["b", "c", "d", "e"],
"d" : ["a", "c"],
"e" : ["c"],
"f" : ["a"]
}
"""
g2 = Graph()
vertices = ['a', 'b', 'c', 'd', 'e', 'f']
for v in vertices:
g2.addVertex(v)
g2.addEdge('a', 'd')
g2.addEdge('a', 'f')
g2.addEdge('b', 'c')
g2.addEdge('c', 'b')
g2.addEdge('c', 'd')
g2.addEdge('d', 'a')
g2.addEdge('d', 'c')
g2.addEdge('c', 'e')
g2.addEdge('e', 'c')
g2.addEdge('f', 'a')
self.assertTrue(g2.isConnected('a'))
"""
g3 = {
"a" : ["d","f"],
"b" : ["c","b"],
"c" : ["b", "c", "d", "e"],
"d" : ["a", "c"],
"e" : ["c"],
"f" : ["a"]
}
"""
g3 = Graph()
vertices = ['a', 'b', 'c', 'd', 'e', 'f']
for v in vertices:
g3.addVertex(v)
g3.addEdge('a', 'd')
g3.addEdge('a', 'f')
g3.addEdge('b', 'c')
g3.addEdge('b', 'b')
g3.addEdge('c', 'b')
g3.addEdge('c', 'c')
g3.addEdge('c', 'd')
g3.addEdge('c', 'e')
g3.addEdge('d', 'a')
g3.addEdge('d', 'c')
g3.addEdge('e', 'c')
g3.addEdge('f', 'a')
self.assertTrue(g3.isConnected('a'))
def testCLRDfs(self):
paths = self.g.CLR_Dfs()
self.assertEqual(len(paths), 1)
self.g.addEdge(1, 0, 2) # add a edge from 1 to 0 to make possible paths 2
paths = self.g.CLR_Dfs()
self.assertEqual(len(paths), 2)
def test_BFS(self):
ss_g = Graph()
for v in range(6):
ss_g.addVertex(v)
ss_g.addEdge(0, 1)
ss_g.addEdge(0, 4)
ss_g.addEdge(0, 5)
ss_g.addEdge(1, 2)
ss_g.addEdge(2, 3)
ss_g.addEdge(3, 4)
(discovered, parents) = ss_g.BFS(0)
expectePath = [0, 1, 2, 3]
vPath = []
ss_g.findPath(0, 3, parents, vPath)
for (x, y) in zip(expectePath, vPath):
self.assertEqual(x, y)
# add an edge between 4 and 3 to short cut for testing new path
ss_g.addEdge(4, 3)
(discovered, parents) = ss_g.BFS(0)
expectePath = [0, 4, 3]
vPath = []
ss_g.findPath(0, 3, parents, vPath)
for (x, y) in zip(expectePath, vPath):
self.assertEqual(x, y)
def testFindPath(self):
g = Graph()
g.addVertex('s')
g.addVertex('u')
g.addVertex('v')
g.addVertex('t')
g.addEdge('s', 'u', 20)
g.addEdge('s', 'v', 10)
g.addEdge('u', 'v', 30)
g.addEdge('u', 't', 10)
g.addEdge('v', 't', 20)
p = g.find_path('s', 't', [])
self.assertTrue(all(x in p for x in ['s', 'u', 't']))
def testFindAllPaths(self):
g = Graph()
g.addVertex('s')
g.addVertex('u')
g.addVertex('v')
g.addVertex('t')
g.addEdge('s', 'u', 20)
g.addEdge('s', 'v', 10)
g.addEdge('u', 'v', 30)
g.addEdge('u', 't', 10)
g.addEdge('v', 't', 20)
paths = g.find_all_paths('s', 't', [])
expectedPaths = [
['s', 'u', 't'],
['s', 'u', 'v', 't'],
['s', 'v', 't']
]
for path in expectedPaths:
self.assertTrue(path in paths)
def testFindShortestPath(self):
g = Graph()
g.addVertex('s')
g.addVertex('u')
g.addVertex('v')
g.addVertex('t')
g.addEdge('s', 'u', 20)
g.addEdge('s', 'v', 10)
g.addEdge('u', 'v', 30)
g.addEdge('u', 't', 10)
g.addEdge('v', 't', 20)
expectedPath = ['s', 'u', 't']
actualPath = g.find_shortest_path('s', 't', [])
for v in expectedPath:
self.assertTrue(v in actualPath)
def testPrims(self):
expectedMST = [(0, 2, 4), (2, 4, 1), (4, 3, 1), (4, 6, 2), (2, 1, 3), (4, 5, 5)]
actualMST = self.g.mspPrims()
for edge in expectedMST:
self.assertTrue(edge in actualMST)
# second example
grph = Graph()
for v in range(8):
grph.addVertex(v)
grph.addEdge(0, 1, 4)
grph.addEdge(0, 2, 6)
grph.addEdge(0, 3, 16)
grph.addEdge(1, 5, 24)
grph.addEdge(2, 5, 23)
grph.addEdge(2, 4, 5)
grph.addEdge(2, 3, 8)
grph.addEdge(3, 4, 10)
grph.addEdge(3, 7, 21)
grph.addEdge(4, 5, 18)
grph.addEdge(4, 6, 11)
grph.addEdge(4, 7, 14)
grph.addEdge(5, 6, 9)
grph.addEdge(6, 7, 7)
expMST = [(0, 1, 4), (0, 2, 6), (2, 4, 5), (2, 3, 8), (4, 6, 11), (6, 7, 7), (6, 5, 9)]
actualMST = grph.mspPrims()
for edge in expMST:
self.assertTrue(edge in actualMST)
# third example
grph = Graph()
for v in range(7):
grph.addVertex(v)
grph.addEdge(0, 1, 4)
grph.addEdge(0, 2, 8)
grph.addEdge(1, 2, 9)
grph.addEdge(1, 3, 8)
grph.addEdge(1, 4, 10)
grph.addEdge(2, 3, 2)
grph.addEdge(2, 5, 1)
grph.addEdge(3, 4, 7)
grph.addEdge(3, 5, 9)
grph.addEdge(4, 5, 5)
grph.addEdge(4, 6, 6)
grph.addEdge(5, 6, 2)
expMST = [(0, 1, 4), (0, 2, 8), (2, 5, 1), (2, 3, 2), (5, 6, 2), (5, 4, 5)]
actualMST = grph.mspPrims()
for edge in expMST:
self.assertTrue(edge in actualMST)
# fourth example
grph = Graph()
for v in range(6):
grph.addVertex(v)
grph.addEdge(0, 1, 2)
grph.addEdge(0, 2, 3)
grph.addEdge(1, 2, 6)
grph.addEdge(1, 3, 5)
grph.addEdge(1, 4, 3)
grph.addEdge(2, 4, 2)
grph.addEdge(3, 4, 1)
grph.addEdge(3, 5, 2)
grph.addEdge(4, 5, 4)
expMST = [(0, 1, 2), (0, 2, 3), (2, 4, 2), (4, 3, 1), (3, 5, 2)]
actualMST = grph.mspPrims()
for edge in expMST:
self.assertTrue(edge in actualMST)
def testKrushkals(self):
expectedMST = [(2, 4, 1), (3, 4, 1), (4, 6, 2), (1, 2, 3), (0, 2, 4), (4, 5, 5)]
actualTree = self.g.mspKrushkals()
for edge in expectedMST:
self.assertTrue(edge in actualTree)
# second example
grph = Graph()
for v in range(7):
grph.addVertex(v)
grph.addEdge(0, 1, 5)
grph.addEdge(0, 2, 7)
grph.addEdge(1, 2, 9)
grph.addEdge(1, 3, 6)
grph.addEdge(1, 5, 15)
grph.addEdge(2, 4, 8)
grph.addEdge(2, 5, 7)
grph.addEdge(3, 5, 8)
grph.addEdge(3, 6, 11)
grph.addEdge(4, 5, 5)
grph.addEdge(5, 6, 9)
expTree = [(0, 1, 5), (4, 5, 5), (1, 3, 6), (0, 2, 7), (2, 5, 7), (5, 6, 9)]
actualTree = grph.mspKrushkals()
for edge in expTree:
self.assertTrue(edge in actualTree)
def testFloyds(self):
expected = {
0: {0: 0, 1: 6, 2: 4, 3: 13, 4: 5, 5: 10, 6: 7 },
1: {0: float('inf'), 1: 0, 2: 3,
3: 12, 4: 4, 5: 7, 6: 6
},
2: {0: float('inf'), 1: float('inf'),
2: 0, 3: 9, 4: 1, 5: 6, 6: 3
},
3: {0: float('inf'), 1: float('inf'), 2: float('inf'),
3: 0, 4: 1, 5: 6, 6: 3
},
4: {0: float('inf'), 1: float('inf'), 2: float('inf'),
3: float('inf'), 4: 0, 5: 5, 6: 2
},
5: {0: float('inf'), 1: float('inf'), 2: float('inf'),
3: float('inf'), 4: float('inf'), 5: 0, 6: float('inf')
},
6: {0: float('inf'), 1: float('inf'), 2: float('inf'),
3: float('inf'), 4: float('inf'), 5: float('inf'), 6: 0
}
}
actual = self.g.floyds()
self.assertDictEqual(expected, actual) # handy
self.assertTrue(p==q for p, q in zip(expected.items(), actual.items())) #does the same
self.assertTrue(expected.items() == actual.items()) #does the same
"""
grph = {0: {0: 0, 1: 1, 2: 4},
1: {0: inf, 1: 0, 2: 2},
2: {0: inf, 1: inf, 2: 0}
}
"""
grph = Graph()
for v in range(3):
grph.addVertex(v)
grph.addEdge(0, 1, 1)
grph.addEdge(0, 2, 4)
grph.addEdge(1, 2, 2)
"""
{0: {0: 0, 1: 1, 2: 3},
1: {0: inf, 1: 0, 2: 2},
2: {0: inf, 1: inf, 2: 0}}
"""
expected = {0: {0: 0, 1: 1, 2: 3},
1: {0: float('inf'), 1: 0, 2: 2},
2: {0: float('inf'), 1: float('inf'), 2: 0}
}
actual = grph.floyds()
self.assertDictEqual(expected, actual) # handy
self.assertTrue(p==q for p, q in zip(expected.items(), actual.items())) #does the same
self.assertTrue(expected.items() == actual.items()) #does the same
def testReachability(self): #uses floyd's algorithm
expected = {
0: {0: True, 1: True, 2: True, 3: True, 4: True, 5: True, 6: True},
1: {0: False, 1: True, 2: True, 3: True, 4: True, 5: True, 6: True},
2: {0: False, 1: False, 2: True, 3: True, 4: True, 5: True, 6: True},
3: {0: False, 1: False, 2: False, 3: True, 4: True, 5: True, 6: True},
4: {0: False, 1: False, 2: False, 3: False, 4: True, 5: True, 6: True},
5: {0: False, 1: False, 2: False, 3: False, 4: False, 5: True, 6: False},
6: {0: False, 1: False, 2: False, 3: False, 4: False, 5: False, 6: True}
}
actual = self.g.reachability()
self.assertDictEqual(expected, actual)
def testPathRecoveryFloydWarshall(self): #ToDo:
expectedParentMap = {
0: {0: -1, 1: 0, 2: 0, 3: 2, 4: 2, 5: 4, 6: 4},
1: {0: -1, 1: -1, 2: 1, 3: 2, 4: 2, 5: 1, 6: 4},
2: {0: -1, 1: -1, 2: -1, 3: 2, 4: 2, 5: 4, 6: 4},
3: {0: -1, 1: -1, 2: -1, 3: -1, 4: 3, 5: 4, 6: 4},
4: {0: -1, 1: -1, 2: -1, 3: -1, 4: -1, 5: 4, 6: 4},
5: {0: -1, 1: -1, 2: -1, 3: -1, 4: -1, 5: -1, 6: -1},
6: {0: -1, 1: -1, 2: -1, 3: -1, 4: -1, 5: -1, 6: -1}
}
actualMap = self.g.pathRecoveryFloydWarshall()
self.assertDictEqual(expectedParentMap, actualMap)
path = []
self.g.getFloydPath(actualMap, 0, 6, path)
self.assertEqual([6, 4, 2, 0], path)
path = []
self.g.getFloydPath(actualMap, 0, 5, path) #always shortest path
self.assertEqual([5, 4, 2, 0], path)
path = []
self.g.getFloydPath(actualMap, 0, 4, path) #always shortest path
self.assertEqual([4, 2, 0], path)
path = []
self.g.getFloydPath(actualMap, 0, 3, path) #always shortest path
self.assertEqual([3, 2, 0], path)
path = []
self.g.getFloydPath(actualMap, 0, 2, path) #always shortest path
self.assertEqual([2, 0], path)
path = []
self.g.getFloydPath(actualMap, 0, 1, path) #always shortest path
self.assertEqual([1, 0], path)
path = []
self.g.getFloydPath(actualMap, 1, 5, path) #always shortest path
self.assertEqual([5, 1], path)
path = []
self.g.getFloydPath(actualMap, 1, 6, path) #always shortest path
self.assertEqual([6, 4, 2, 1], path)
path = []
self.g.getFloydPath(actualMap, 2, 6, path) #always shortest path
self.assertEqual([6, 4, 2], path)
path = []
self.g.getFloydPath(actualMap, 2, 5, path) #always shortest path
self.assertEqual([5, 4, 2], path)
def testDijkstra(self):
g = Graph()
g.addVertex('s')
g.addVertex('u')
g.addVertex('v')
g.addVertex('x')
g.addVertex('y')
g.addEdge('s', 'u', 10)
g.addEdge('s', 'x', 5)
g.addEdge('u', 'v', 1)
g.addEdge('u', 'x', 2)
g.addEdge('v', 'y', 4)
g.addEdge('x', 'u', 3)
g.addEdge('x', 'v', 9)
g.addEdge('x', 'y', 2)
g.addEdge('y', 's', 7)
g.addEdge('y', 'v', 6)
expected = ['y', 's']
actual = g.shortestPathDijkstra('y', 's')
self.assertEqual(expected, actual)
expected = ['s', 'x', 'u', 'v']
actual = g.shortestPathDijkstra('s', 'v')
self.assertEqual(expected, actual)
g.updateEdge('s', 'u', 1)
expected = ['s', 'u', 'v']
actual = g.shortestPathDijkstra('s', 'v')
self.assertEqual(expected, actual)
def testTarjansSCC(self):
expected = [('G',), ('B',), ('E', 'F', 'D'), ('C',), ('A',)]
g = Graph()
g.addVertex('A')
g.addVertex('B')
g.addVertex('C')
g.addVertex('D')
g.addVertex('E')
g.addVertex('F')
g.addVertex('G')
g.addEdge('A', 'C', 1)
g.addEdge('A', 'E', 1)
g.addEdge('A', 'F', 1)
g.addEdge('B', 'G', 1)
g.addEdge('C', 'B', 1)
g.addEdge('C', 'D', 1)
g.addEdge('F', 'E', 1)
g.addEdge('D', 'F', 1)
g.addEdge('E', 'D', 1)
g.addEdge('E', 'G', 1)
self.assertEqual(expected, g.strongly_connected_components())
#case -2
expected = [(12, 11, 9, 10, 7, 8), (6, 3), (5, 4, 2), (1,)]
g2 = Graph()
for v in range(1, 13):
g2.addVertex(v)
g2.addEdge(1, 2, 1)
g2.addEdge(2, 3, 1)
g2.addEdge(2, 4, 1)
g2.addEdge(2, 5, 1)
g2.addEdge(3, 6, 1)
g2.addEdge(4, 5, 1)
g2.addEdge(4, 7, 1)
g2.addEdge(5, 2, 1)
g2.addEdge(5, 6, 1)
g2.addEdge(5, 7, 1)
g2.addEdge(6, 3, 1)
g2.addEdge(6, 8, 1)
g2.addEdge(7, 8, 1)
g2.addEdge(7, 10, 1)
g2.addEdge(8, 7, 1)
g2.addEdge(9, 7, 1)
g2.addEdge(10, 9, 1)
g2.addEdge(10, 11, 1)
g2.addEdge(11, 12, 1)
g2.addEdge(12, 10, 1)
self.assertEqual(expected, g2.strongly_connected_components())
"""
def testIsConnected(self):
self.assertTrue(self.g.isConnected())
self.g.addVertex(10)
self.assertFalse(self.g.isConnected())
"""
g = { "a" : ["d"],
"b" : ["c"],
"c" : ["b", "c", "d", "e"],
"d" : ["a", "c"],
"e" : ["c"],
"f" : []
}
"""
g = Graph()
vertices = ['a', 'b', 'c', 'd', 'e', 'f']
for v in vertices:
g.addVertex(v)
g.addEdge('a', 'd')
g.addEdge('b', 'c')
g.addEdge('c', 'd')
g.addEdge('c', 'e')
self.assertFalse(g.isConnected('a'))
"""
g2 = {
"a" : ["d","f"],
"b" : ["c"],
"c" : ["b", "c", "d", "e"],
"d" : ["a", "c"],
"e" : ["c"],
"f" : ["a"]
}
"""
g2 = Graph()
vertices = ['a', 'b', 'c', 'd', 'e', 'f']
for v in vertices:
g2.addVertex(v)
g2.addEdge('a', 'd')
g2.addEdge('a', 'f')
g2.addEdge('b', 'c')
g2.addEdge('c', 'b')
g2.addEdge('c', 'd')
g2.addEdge('d', 'a')
g2.addEdge('d', 'c')
g2.addEdge('c', 'e')
g2.addEdge('e', 'c')
g2.addEdge('f', 'a')
self.assertTrue(g2.isConnected('a'))
"""
g3 = {
"a" : ["d","f"],
"b" : ["c","b"],
"c" : ["b", "c", "d", "e"],
"d" : ["a", "c"],
"e" : ["c"],
"f" : ["a"]
}
"""
g3 = Graph()
vertices = ['a', 'b', 'c', 'd', 'e', 'f']
for v in vertices:
g3.addVertex(v)
g3.addEdge('a', 'd')
g3.addEdge('a', 'f')
g3.addEdge('b', 'c')
g3.addEdge('b', 'b')
g3.addEdge('c', 'b')
g3.addEdge('c', 'c')
g3.addEdge('c', 'd')
g3.addEdge('c', 'e')
g3.addEdge('d', 'a')
g3.addEdge('d', 'c')
g3.addEdge('e', 'c')
g3.addEdge('f', 'a')
self.assertTrue(g3.isConnected('a'))