def construct(self):
        time_per_branch = 0.5
        text = TextMobject(
            """
            In any tree:
            $$E + 1 = V$$
        """
        )
        gs = GraphScene(SampleGraph())
        gs.generate_treeified_spanning_tree()
        branches = gs.treeified_spanning_tree.to_edge(LEFT).split()

        all_dots = [Dot(branches[0].points[0])]
        self.add(text, all_dots[0])
        for branch in branches:
            self.play(ShowCreation(branch), run_time=time_per_branch)
            dot = Dot(branch.points[-1])
            self.add(dot)
            all_dots.append(dot)
        self.dither()
        self.remove(*all_dots)
        self.play(
            FadeOut(text),
            FadeIn(Mobject(*gs.edges + gs.vertices)),
            *[Transform(*pair) for pair in zip(branches, gs.spanning_tree.split())]
        )

class GraphWidget(QtGui.QGraphicsView):
    
    def __init__(self, index, parent=None):
        super(GraphWidget, self).__init__(parent)
        self.index = index
        self.createWidgets()
        self.updateWidgets()
        
    def mouseMoveEvent(self, event):
        super(GraphWidget, self).mouseMoveEvent(event)
        pt = event.pos()
        self.statusBarShowMessage('Coordinate (%2.f, %2.f)' % (pt.x(), pt.y()))
    
    def createWidgets(self):
        self.scene = GraphScene(self)
        self.scene.setItemIndexMethod(QtGui.QGraphicsScene.NoIndex)
        self.scene.setSceneRect(-200, -200, 400, 400)
        
    def updateWidgets(self):
        self.setObjectName('GraphWidget-%d' % self.index)
        self.setScene(self.scene)
        self.setRenderHint(QtGui.QPainter.Antialiasing)
        self.scale(0.8, 0.8)
        self.setMinimumSize(400, 400)
        self.setWindowTitle(
            QtGui.QApplication.translate('GraphWidget-%d' % self.index, 
                                         'Polygon Area', 
                                         None,
                                         QtGui.QApplication.UnicodeUTF8))
        self.setMouseTracking(True)
        
    def statusBarShowMessage(self, st):
        qst = QtCore.QString(st)
        mainWindow = QtGui.qApp.mainWindow
        mainWindow.statusBar().showMessage(qst)

    def construct(self):
        GraphScene.construct(self)
        self.generate_dual_graph()

        self.add(TextMobject("Duality").to_edge(UP))
        self.remove(*self.vertices)
        def special_alpha(t):
            if t > 0.5:
                t = 1 - t
            if t < 0.25:
                return smooth(4*t)
            else:
                return 1
        kwargs = {
            "run_time" : 5.0,
            "rate_func" : special_alpha
        }
        self.play(*[
            Transform(*edge_pair, **kwargs)
            for edge_pair in zip(self.edges, self.dual_edges)
        ] + [
            Transform(
                Mobject(*[
                    self.vertices[index]
                    for index in cycle
                ]),
                dv,
                **kwargs
            )
            for cycle, dv in zip(
                self.graph.region_cycles, 
                self.dual_vertices
            )
        ])
        self.dither()

 def createWidgets(self):
     self.scene = GraphScene(self)
     self.scene.setItemIndexMethod(QtGui.QGraphicsScene.NoIndex)
     self.scene.setSceneRect(-200, -200, 400, 400)

 def __init__(self, graph, path, *args, **kwargs):
     self.path = path
     GraphScene.__init__(self, graph, *args, **kwargs)