# Graph Plotting¶

(For LaTeX drawings of graphs, see the graph_latex module.)

All graphs have an associated Sage graphics object, which you can display:

sage: G = graphs.WheelGraph(15)
sage: P = G.plot()
sage: P.show() # long time


If you create a graph in Sage using the Graph command, then plot that graph, the positioning of nodes is determined using the spring-layout algorithm. For the special graph constructors, which you get using graphs.[tab], the positions are preset. For example, consider the Petersen graph with default node positioning vs. the Petersen graph constructed by this database:

sage: petersen_spring = Graph({0:[1,4,5], 1:[0,2,6], 2:[1,3,7], 3:[2,4,8], 4:[0,3,9], 5:[0,7,8], 6:[1,8,9], 7:[2,5,9], 8:[3,5,6], 9:[4,6,7]})
sage: petersen_spring.show() # long time
sage: petersen_database = graphs.PetersenGraph()
sage: petersen_database.show() # long time


For all the constructors in this database (except the octahedral, dodecahedral, random and empty graphs), the position dictionary is filled in, instead of using the spring-layout algorithm.

Plot options

Here is the list of options accepted by plot() and the constructor of GraphPlot.

 partition A partition of the vertex set. If specified, plot will show each cell in a different color. vertex_colors takes precedence. dist The distance between multiedges. vertex_labels Whether or not to draw vertex labels. edge_color The default color for edges. spring Use spring layout to finalize the current layout. pos The position dictionary of vertices loop_size The radius of the smallest loop. color_by_label Whether to color the edges according to their labels. This also accepts a function or dictionary mapping labels to colors. iterations The number of times to execute the spring layout algorithm. talk Whether to display the vertices in talk mode (larger and white). edge_labels Whether or not to draw edge labels. vertex_size The size to draw the vertices. dim The dimension of the layout – 2 or 3. edge_style The linestyle of the edges. It should be one of “solid”, “dashed”, “dotted”, dashdot”, or “-”, “–”, ”:”, “-.”, respectively. This currently only works for directed graphs, since we pass off the undirected graph to networkx. layout A layout algorithm – one of : “acyclic”, “circular” (plots the graph with vertices evenly distributed on a circle), “ranked”, “graphviz”, “planar”, “spring” (traditional spring layout, using the graph’s current positions as initial positions), or “tree” (the tree will be plotted in levels, depending on minimum distance for the root). vertex_shape The shape to draw the vertices. Currently unavailable for Multi-edged DiGraphs. vertex_colors Dictionary of vertex coloring : each key is a color recognizable by matplotlib, and each corresponding entry is a list of vertices. If a vertex is not listed, it looks invisible on the resulting plot (it does not get drawn). by_component Whether to do the spring layout by connected component – a boolean. heights A dictionary mapping heights to the list of vertices at this height. graph_border Whether or not to draw a frame around the graph. max_dist The max distance range to allow multiedges. prog Which graphviz layout program to use – one of “circo”, “dot”, “fdp”, “neato”, or “twopi”. edge_colors a dictionary specifying edge colors: each key is a color recognized by matplotlib, and each entry is a list of edges. tree_orientation The direction of tree branches – ‘up’, ‘down’, ‘left’ or ‘right’. save_pos Whether or not to save the computed position for the graph. tree_root A vertex designation for drawing trees. A vertex of the tree to be used as the root for the layout='tree' option. If no root is specified, then one is chosen close to the center of the tree. Ignored unless layout='tree'

Default options

This module defines two dictionaries containing default options for the plot() and show() methods. These two dictionaries are sage.graphs.graph_plot.DEFAULT_PLOT_OPTIONS and sage.graphs.graph_plot.DEFAULT_SHOW_OPTIONS, respectively.

Obviously, these values are overruled when arguments are given explicitly.

Here is how to define the default size of a graph drawing to be [6,6]. The first two calls to show() use this option, while the third does not (a value for figsize is explicitly given):

sage: sage.graphs.graph_plot.DEFAULT_SHOW_OPTIONS['figsize'] = [6,6]
sage: graphs.PetersenGraph().show() # long time
sage: graphs.ChvatalGraph().show()  # long time
sage: graphs.PetersenGraph().show(figsize=[4,4]) # long time


We can now reset the default to its initial value, and now display graphs as previously:

sage: sage.graphs.graph_plot.DEFAULT_SHOW_OPTIONS['figsize'] = [4,4]
sage: graphs.PetersenGraph().show() # long time
sage: graphs.ChvatalGraph().show()  # long time


Note

• While DEFAULT_PLOT_OPTIONS affects both G.show() and G.plot(), settings from DEFAULT_SHOW_OPTIONS only affects G.show().

• In order to define a default value permanently, you can add a couple of lines to Sage’s startup scripts. Example

sage: import sage.graphs.graph_plot
sage: sage.graphs.graph_plot.DEFAULT_SHOW_OPTIONS['figsize'] = [4,4]


Index of methods and functions

 GraphPlot.set_pos() Sets the position plotting parameters for this GraphPlot. GraphPlot.set_vertices() Sets the vertex plotting parameters for this GraphPlot. GraphPlot.set_edges() Sets the edge (or arrow) plotting parameters for the GraphPlot object. GraphPlot.show() Shows the (Di)Graph associated with this GraphPlot object. GraphPlot.plot() Returns a graphics object representing the (di)graph. GraphPlot.layout_tree() Compute a nice layout of a tree. _circle_embedding() Sets some vertices on a circle in the embedding of a graph G. _line_embedding() Sets some vertices on a line in the embedding of a graph G.

## Methods and classes¶

sage.graphs.graph_plot._circle_embedding(g, vertices, center=(0, 0), radius=1, shift=0)

Sets some vertices on a circle in the embedding of a graph G.

This method modifies the graph’s embedding so that the vertices listed in vertices appear in this ordering on a circle of given radius and center. The shift parameter is actually a rotation of the circle. A value of shift=1 will replace in the drawing the $$i$$-th element of the list by the $$(i-1)$$-th. Non-integer values are admissible, and a value of $$\alpha$$ corresponds to a rotation of the circle by an angle of $$\alpha 2\pi/n$$ (where $$n$$ is the number of vertices set on the circle).

EXAMPLE:

sage: from sage.graphs.graph_plot import _circle_embedding
sage: g = graphs.CycleGraph(5)
sage: _circle_embedding(g, [0, 2, 4, 1, 3], radius=2, shift=.5)
sage: g.show()

sage.graphs.graph_plot._line_embedding(g, vertices, first=(0, 0), last=(0, 1))

Sets some vertices on a line in the embedding of a graph G.

This method modifies the graph’s embedding so that the vertices of vertices appear on a line, where the position of vertices[0] is the pair first and the position of vertices[-1] is last. The vertices are evenly spaced.

EXAMPLE:

sage: from sage.graphs.graph_plot import _line_embedding
sage: g = graphs.PathGraph(5)
sage: _line_embedding(g, [0, 2, 4, 1, 3], first=(-1, -1), last=(1, 1))
sage: g.show()

class sage.graphs.graph_plot.GraphPlot(graph, options)

Returns a GraphPlot object, which stores all the parameters needed for plotting (Di)Graphs. A GraphPlot has a plot and show function, as well as some functions to set parameters for vertices and edges. This constructor assumes default options are set. Defaults are shown in the example below.

EXAMPLE:

sage: from sage.graphs.graph_plot import GraphPlot
sage: options = {
...     'vertex_size':200,
...     'vertex_labels':True,
...     'layout':None,
...     'edge_style':'solid',
...     'edge_color':'black',
...     'edge_colors':None,
...     'edge_labels':False,
...     'iterations':50,
...     'tree_orientation':'down',
...     'heights':None,
...     'graph_border':False,
...     'talk':False,
...     'color_by_label':False,
...     'partition':None,
...     'dist':.075,
...     'max_dist':1.5,
...     'loop_size':.075}
sage: g = Graph({0:[1,2], 2:[3], 4:[0,1]})
sage: GP = GraphPlot(g, options)

layout_tree(root, orientation)

Compute a nice layout of a tree.

INPUT:

• root – the root vertex.

• orientation – Whether to place the root at the top or at the bottom :

• orientation="down" – children are placed below their parent
• orientation="top" – children are placed above their parent

EXAMPLES:

sage: T = graphs.RandomLobster(25,0.3,0.3)
sage: T.show(layout='tree',tree_orientation='up') # indirect doctest

sage: from sage.graphs.graph_plot import GraphPlot
sage: G = graphs.HoffmanSingletonGraph()
sage: T = Graph()
sage: T.show(layout='tree',tree_root=0) # indirect doctest

plot(**kwds)

Returns a graphics object representing the (di)graph.

INPUT:

The options accepted by this method are to be found in the documentation of the sage.graphs.graph_plot module, and the show() method.

Note

See the module's documentation for information on default values of this method.

We can specify some pretty precise plotting of familiar graphs:

sage: from math import sin, cos, pi
sage: P = graphs.PetersenGraph()
sage: d = {'#FF0000':[0,5], '#FF9900':[1,6], '#FFFF00':[2,7], '#00FF00':[3,8], '#0000FF':[4,9]}
sage: pos_dict = {}
sage: for i in range(5):
...    x = float(cos(pi/2 + ((2*pi)/5)*i))
...    y = float(sin(pi/2 + ((2*pi)/5)*i))
...    pos_dict[i] = [x,y]
...
sage: for i in range(10)[5:]:
...    x = float(0.5*cos(pi/2 + ((2*pi)/5)*i))
...    y = float(0.5*sin(pi/2 + ((2*pi)/5)*i))
...    pos_dict[i] = [x,y]
...
sage: pl = P.graphplot(pos=pos_dict, vertex_colors=d)
sage: pl.show()


Here are some more common graphs with typical options:

sage: C = graphs.CubeGraph(8)
sage: P = C.graphplot(vertex_labels=False, vertex_size=0, graph_border=True)
sage: P.show()

sage: G = graphs.HeawoodGraph().copy(sparse=True)
sage: for u,v,l in G.edges():
...    G.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')')
sage: G.graphplot(edge_labels=True).show()


The options for plotting also work with directed graphs:

sage: D = DiGraph( { 0: [1, 10, 19], 1: [8, 2], 2: [3, 6], 3: [19, 4], 4: [17, 5], 5: [6, 15], 6: [7], 7: [8, 14], 8: [9], 9: [10, 13], 10: [11], 11: [12, 18], 12: [16, 13], 13: [14], 14: [15], 15: [16], 16: [17], 17: [18], 18: [19], 19: []}, implementation='networkx' )
sage: for u,v,l in D.edges():
...    D.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')')
sage: D.graphplot(edge_labels=True, layout='circular').show()


This example shows off the coloring of edges:

sage: from sage.plot.colors import rainbow
sage: C = graphs.CubeGraph(5)
sage: R = rainbow(5)
sage: edge_colors = {}
sage: for i in range(5):
...    edge_colors[R[i]] = []
sage: for u,v,l in C.edges():
...    for i in range(5):
...        if u[i] != v[i]:
...            edge_colors[R[i]].append((u,v,l))
sage: C.graphplot(vertex_labels=False, vertex_size=0, edge_colors=edge_colors).show()


With the partition option, we can separate out same-color groups of vertices:

sage: D = graphs.DodecahedralGraph()
sage: Pi = [[6,5,15,14,7],[16,13,8,2,4],[12,17,9,3,1],[0,19,18,10,11]]
sage: D.show(partition=Pi)


Loops are also plotted correctly:

sage: G = graphs.PetersenGraph()
sage: G.allow_loops(True)
sage: G.show()

sage: D = DiGraph({0:[0,1], 1:[2], 2:[3]}, loops=True)
sage: D.show()
sage: D.show(edge_colors={(0,1,0):[(0,1,None),(1,2,None)],(0,0,0):[(2,3,None)]})


More options:

sage: pos = {0:[0.0, 1.5], 1:[-0.8, 0.3], 2:[-0.6, -0.8], 3:[0.6, -0.8], 4:[0.8, 0.3]}
sage: g = Graph({0:[1], 1:[2], 2:[3], 3:[4], 4:[0]})
sage: g.graphplot(pos=pos, layout='spring', iterations=0).plot()

sage: G = Graph()
sage: P = G.graphplot().plot()
sage: P.axes()
False
sage: G = DiGraph()
sage: P = G.graphplot().plot()
sage: P.axes()
False


We can plot multiple graphs:

sage: T = list(graphs.trees(7))
sage: t = T[3]
sage: t.graphplot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}).plot()

sage: T = list(graphs.trees(7))
sage: t = T[3]
sage: t.graphplot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}).plot()
sage: t.set_edge_label(0,1,-7)
sage: t.set_edge_label(0,5,3)
sage: t.set_edge_label(0,5,99)
sage: t.set_edge_label(1,2,1000)
sage: t.set_edge_label(3,2,'spam')
sage: t.set_edge_label(2,6,3/2)
sage: t.set_edge_label(0,4,66)
sage: t.graphplot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}, edge_labels=True).plot()

sage: T = list(graphs.trees(7))
sage: t = T[3]
sage: t.graphplot(layout='tree').show()


The tree layout is also useful:

sage: t = DiGraph('JCC???@A??GO??CO??GO??')
sage: t.graphplot(layout='tree', tree_root=0, tree_orientation="up").show()


More examples:

sage: D = DiGraph({0:[1,2,3], 2:[1,4], 3:[0]})
sage: D.graphplot().show()

sage: D = DiGraph(multiedges=True, sparse=True)
sage: for i in range(5):
sage: D.graphplot(edge_labels=True,edge_colors=D._color_by_label()).plot()

sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
...     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: g.graphplot(edge_labels=True, color_by_label=True, edge_style='dashed').plot()


The edge_style option may be provided in the short format too:

sage: g.graphplot(edge_labels=True, color_by_label=True, edge_style='--').plot()


TESTS:

Make sure that show options work with plot also:

sage: g = Graph({})
sage: g.plot(title='empty graph', axes=True)


Check for invalid inputs:

sage: p = graphs.PetersenGraph().plot(egabrag='garbage')
Traceback (most recent call last):
...
ValueError: Invalid input 'egabrag=garbage'

set_edges(**edge_options)

Sets the edge (or arrow) plotting parameters for the GraphPlot object.

This function is called by the constructor but can also be called to make updates to the vertex options of an existing GraphPlot object. Note that the changes are cumulative.

EXAMPLES:

sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
...     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = g.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.set_edges(edge_style='solid')
sage: GP.plot()
sage: GP.set_edges(edge_color='black')
sage: GP.plot()

sage: d = DiGraph({}, loops=True, multiedges=True, sparse=True)
...     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = d.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.set_edges(edge_style='solid')
sage: GP.plot()
sage: GP.set_edges(edge_color='black')
sage: GP.plot()


TESTS:

sage: G = Graph("Fooba")
sage: G.show(edge_colors={'red':[(3,6),(2,5)]})


Verify that default edge labels are pretty close to being between the vertices in some cases where they weren’t due to truncating division (trac ticket #10124):

sage: test_graphs = graphs.FruchtGraph(), graphs.BullGraph()
sage: tol = 0.001
sage: for G in test_graphs:
...       E=G.edges()
...       for e0, e1, elab in E:
...           G.set_edge_label(e0, e1, '%d %d' % (e0, e1))
...       gp = G.graphplot(save_pos=True,edge_labels=True)
...       vx = gp._plot_components['vertices'][0].xdata
...       vy = gp._plot_components['vertices'][0].ydata
...       for elab in gp._plot_components['edge_labels']:
...           textobj = elab[0]
...           x, y, s = textobj.x, textobj.y, textobj.string
...           v0, v1 = map(int, s.split())
...           vn = vector(((x-(vx[v0]+vx[v1])/2.),y-(vy[v0]+vy[v1])/2.)).norm()
...           assert vn < tol

set_pos()

Sets the position plotting parameters for this GraphPlot.

EXAMPLES:

This function is called implicitly by the code below:

sage: g = Graph({0:[1,2], 2:[3], 4:[0,1]})
sage: g.graphplot(save_pos=True, layout='circular') # indirect doctest
GraphPlot object for Graph on 5 vertices


The following illustrates the format of a position dictionary, but due to numerical noise we do not check the values themselves:

sage: g.get_pos()
{0: [...e-17, 1.0],
1: [-0.951..., 0.309...],
2: [-0.587..., -0.809...],
3: [0.587..., -0.809...],
4: [0.951..., 0.309...]}

sage: T = list(graphs.trees(7))
sage: t = T[3]
sage: t.plot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]})


TESTS:

Make sure that vertex locations are floats. Not being floats isn’t a bug in itself but makes it too easy to accidentally introduce a bug elsewhere, such as in set_edges() (trac ticket #10124), via silent truncating division of integers:

sage: g = graphs.FruchtGraph()
sage: gp = g.graphplot()
sage: set(map(type, flatten(gp._pos.values())))
set([<type 'float'>])
sage: g = graphs.BullGraph()
sage: gp = g.graphplot(save_pos=True)
sage: set(map(type, flatten(gp._pos.values())))
set([<type 'float'>])

set_vertices(**vertex_options)

Sets the vertex plotting parameters for this GraphPlot. This function is called by the constructor but can also be called to make updates to the vertex options of an existing GraphPlot object. Note that the changes are cumulative.

EXAMPLES:

sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
...     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = g.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.set_vertices(talk=True)
sage: GP.plot()
sage: GP.set_vertices(vertex_colors='pink', vertex_shape='^')
sage: GP.plot()

show(**kwds)

Shows the (Di)Graph associated with this GraphPlot object.

INPUT:

This method accepts all parameters of sage.plot.graphics.Graphics.show().

Note

EXAMPLE:

sage: C = graphs.CubeGraph(8)
sage: P = C.graphplot(vertex_labels=False, vertex_size=0, graph_border=True)
sage: P.show()