graph_tool.topology - Topology related functions

Summary

shortest_distance (g[, source, weights, ...])	Calculate the distance of all vertices from a given source, or the all pairs shortest paths, if the source is not specified.
shortest_path (g, source, target[, weights, pred_map])	Return the shortest path from source to target.
isomorphism (g1, g2[, isomap])	Check whether two graphs are isomorphic.
subgraph_isomorphism (sub, g)	Obtain all subgraph isomorphisms of sub in g.
mark_subgraph (g, sub, vmap, emap[, vmask, emask])	Mark a given subgraph sub on the graph g.
min_spanning_tree (g[, weights, root, tree_map])	Return the minimum spanning tree of a given graph.
dominator_tree (g, root[, dom_map])	Return a vertex property map the dominator vertices for each vertex.
topological_sort (g)	Return the topological sort of the given graph. It is returned as an array of vertex indexes, in the sort order.
transitive_closure (g)	Return the transitive closure graph of g.
label_components (g[, vprop, directed])	Label the components to which each vertex in the graph belongs. If the graph is directed, it finds the strongly connected components.
label_biconnected_components (g[, eprop, vprop])	Label the edges of biconnected components, and the vertices which are articulation points.
is_planar (g[, embedding, kuratowski])	Test if the graph is planar.

Contents

graph_tool.topology.isomorphism(g1, g2, isomap=False)

Check whether two graphs are isomorphic.

If isomap is True, a vertex PropertyMap with the isomorphism mapping is returned as well.

Examples

>>> from numpy.random import seed
>>> seed(42)
>>> g = gt.random_graph(100, lambda: (3,3))
>>> g2 = gt.Graph(g)
>>> gt.isomorphism(g, g2)
True
>>> g.add_edge(g.vertex(0), g.vertex(1))
<...>
>>> gt.isomorphism(g, g2)
False

graph_tool.topology.subgraph_isomorphism(sub, g)

Obtain all subgraph isomorphisms of sub in g.

It returns two lists, containing the vertex and edge property maps for sub with the isomorphism mappings. The value of the properties are the vertex/edge index of the corresponding vertex/edge in g.

Notes

The algorithm used is described in [ullmann-algorithm-1976]. It has worse-case complexity of O(N_g^{N_{sub}}), but for random graphs it typically has a complexity of O(N_g^\gamma) with \gamma depending sub-linearly on the size of sub.

References

[ullmann-algorithm-1976]

Ullmann, J. R., “An algorithm for subgraph isomorphism”, Journal of the ACM 23 (1): 31–42, 1976, doi:10.1145/321921.321925

[subgraph-isormophism-wikipedia]

http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem

Examples

>>> from numpy.random import seed, poisson
>>> seed(42)
>>> g = gt.random_graph(30, lambda: (poisson(6),poisson(6)))
>>> sub = gt.random_graph(10, lambda: (poisson(1.8), poisson(1.9)))
>>> vm, em = gt.subgraph_isomorphism(sub, g)
>>> print len(vm)
175
>>> for i in xrange(len(vm)):
...   g.set_vertex_filter(None)
...   g.set_edge_filter(None)
...   vmask, emask = gt.mark_subgraph(g, sub, vm[i], em[i])
...   g.set_vertex_filter(vmask)
...   g.set_edge_filter(emask)
...   assert(gt.isomorphism(g, sub))
>>> g.set_vertex_filter(None)
>>> g.set_edge_filter(None)
>>> ewidth = g.copy_property(emask, value_type="double")
>>> ewidth.a *= 1.5
>>> ewidth.a += 0.5
>>> gt.graph_draw(g, vcolor=vmask, ecolor=emask, penwidth=ewidth,
...               output="subgraph-iso-embed.png")
<...>
>>> gt.graph_draw(sub, output="subgraph-iso.png")
<...>

Left: Subgraph searched, Right: One isomorphic subgraph found in main

graph.

graph_tool.topology.mark_subgraph(g, sub, vmap, emap, vmask=None, emask=None)

Mark a given subgraph sub on the graph g.

The mapping must be provided by the vmap and emap parameters, which map vertices/edges of sub to indexes of the corresponding vertices/edges in g.

This returns a vertex and an edge property map, with value type ‘bool’, indicating whether or not a vertex/edge in g corresponds to the subgraph sub.

graph_tool.topology.min_spanning_tree(g, weights=None, root=None, tree_map=None)

Return the minimum spanning tree of a given graph.

Parameters:

g : Graph

Graph to be used.

weights : PropertyMap (optional, default: None)

The edge weights. If provided, the minimum spanning tree will minimize the edge weights.

root : Vertex (optional, default: None)

Root of the minimum spanning tree. It this is provided, Prim’s algorithm is used. Otherwise, Kruskal’s algorithm is used.

tree_map : PropertyMap (optional, default: None)

If provided, the edge tree map will be written in this property map.

Returns:

tree_map : PropertyMap

Edge property map with mark the tree edges: 1 for tree edge, 0 otherwise.

Notes

The algorithm runs with O(E\log E) complexity, or O(E\log V) if root is specified.

References

[kruskal-shortest-1956]

J. B. Kruskal. “On the shortest spanning subtree of a graph and the traveling salesman problem”, In Proceedings of the American Mathematical Sofiety, volume 7, pages 48-50, 1956.

[prim-shortest-1957]

R. Prim. “Shortest connection networks and some generalizations”, Bell System Technical Journal, 36:1389-1401, 1957.

[boost-mst]

http://www.boost.org/libs/graph/doc/graph_theory_review.html#sec:minimum-spanning-tree

[mst-wiki]

http://en.wikipedia.org/wiki/Minimum_spanning_tree

Examples

>>> from numpy.random import seed
>>> seed(42)
>>> g = gt.random_graph(100, lambda: (5, 5))
>>> tree = gt.min_spanning_tree(g)
>>> print tree.a
[0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0
 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1
 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0
 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1
 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0
 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0
 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0
 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0
 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0
 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0]

graph_tool.topology.dominator_tree(g, root, dom_map=None)

Return a vertex property map the dominator vertices for each vertex.

Parameters:

g : Graph

Graph to be used.

root : Vertex

The root vertex.

dom_map : PropertyMap (optional, default: None)

If provided, the dominator map will be written in this property map.

Returns:

dom_map : PropertyMap

The dominator map. It contains for each vertex, the index of its dominator vertex.

Notes

A vertex u dominates a vertex v, if every path of directed graph from the entry to v must go through u.

The algorithm runs with O((V+E)\log (V+E)) complexity.

References

[dominator-bgl]

http://www.boost.org/libs/graph/doc/lengauer_tarjan_dominator.htm

Examples

>>> from numpy.random import seed
>>> seed(42)
>>> g = gt.random_graph(100, lambda: (2, 2))
>>> tree = gt.min_spanning_tree(g)
>>> g.set_edge_filter(tree)
>>> root = [v for v in g.vertices() if v.in_degree() == 0]
>>> dom = gt.dominator_tree(g, root[0])
>>> print dom.a
[ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  3  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  0  0  0  0 20  0  0  0  0  0  0  0  0  0  0
 78  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 26  0  0
  0  0  0 20  0  0  0  0  0  0  0 50  0  0  0  0  0  0  0  0  0  0  0  0  0]

graph_tool.topology.topological_sort(g)

Return the topological sort of the given graph. It is returned as an array of vertex indexes, in the sort order.

Notes

The topological sort algorithm creates a linear ordering of the vertices such that if edge (u,v) appears in the graph, then v comes before u in the ordering. The graph must be a directed acyclic graph (DAG).

The time complexity is O(V + E).

References

[topological-boost]

http://www.boost.org/libs/graph/doc/topological_sort.html

[topological-wiki]

http://en.wikipedia.org/wiki/Topological_sorting

Examples

>>> from numpy.random import seed
>>> seed(42)
>>> g = gt.random_graph(30, lambda: (3, 3))
>>> tree = gt.min_spanning_tree(g)
>>> g.set_edge_filter(tree)
>>> sort = gt.topological_sort(g)
>>> print sort
[25 24  0 15  5 14  4  7 19  9 20  3 12 28 22 16  1 13  8  2  6 27 10 29 21
 11 17 18 23 26]

graph_tool.topology.transitive_closure(g)

Return the transitive closure graph of g.

Notes

The transitive closure of a graph G = (V,E) is a graph G* = (V,E*) such that E* contains an edge (u,v) if and only if G contains a path (of at least one edge) from u to v. The transitive_closure() function transforms the input graph g into the transitive closure graph tc.

The time complexity (worst-case) is O(VE).

References

[transitive-boost]

http://www.boost.org/libs/graph/doc/transitive_closure.html

[transitive-wiki]

http://en.wikipedia.org/wiki/Transitive_closure

Examples

>>> from numpy.random import seed
>>> seed(42)
>>> g = gt.random_graph(30, lambda: (3, 3))
>>> tc = gt.transitive_closure(g)

graph_tool.topology.label_components(g, vprop=None, directed=None)

Label the components to which each vertex in the graph belongs. If the graph is directed, it finds the strongly connected components.

Parameters:

g : Graph

Graph to be used.

vprop : PropertyMap (optional, default: None)

Vertex property to store the component labels. If none is supplied, one is created.

directed : bool (optional, default:None)

Treat graph as directed or not, independently of its actual directionality.

Returns:

comp : PropertyMap

Vertex property map with component labels.

Notes

The components are arbitrarily labeled from 0 to N-1, where N is the total number of components.

The algorithm runs in O(V + E) time.

Examples

>>> from numpy.random import seed
>>> seed(43)
>>> g = gt.random_graph(100, lambda: (1, 1))
>>> comp = gt.label_components(g)
>>> print comp.get_array()
[0 0 0 0 1 0 2 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 2 0 1 0 0 1 0 0 2 0 0 0
 2 2 0 0 0 2 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 1 0 0 2 2 0 1 0 1 0 0 0 0 0 1 1
 0 2 1 0 0 2 0 0 2 0 3 2 0 0 3 2 0 0 0 0 0 0 1 1 1 0]

graph_tool.topology.label_biconnected_components(g, eprop=None, vprop=None)

Label the edges of biconnected components, and the vertices which are articulation points.

Parameters:

g : Graph

Graph to be used.

eprop : PropertyMap (optional, default: None)

Edge property to label the biconnected components.

vprop : PropertyMap (optional, default: None)

Vertex property to mark the articulation points. If none is supplied, one is created.

Returns:

bicomp : PropertyMap

Edge property map with the biconnected component labels.

articulation : PropertyMap

Boolean vertex property map which has value 1 for each vertex which is an articulation point, and zero otherwise.

nc : int

Number of biconnected components.

Notes

A connected graph is biconnected if the removal of any single vertex (and all edges incident on that vertex) can not disconnect the graph. More generally, the biconnected components of a graph are the maximal subsets of vertices such that the removal of a vertex from a particular component will not disconnect the component. Unlike connected components, vertices may belong to multiple biconnected components: those vertices that belong to more than one biconnected component are called “articulation points” or, equivalently, “cut vertices”. Articulation points are vertices whose removal would increase the number of connected components in the graph. Thus, a graph without articulation points is biconnected. Vertices can be present in multiple biconnected components, but each edge can only be contained in a single biconnected component.

The algorithm runs in O(V + E) time.

Examples

>>> from numpy.random import seed
>>> seed(42)
>>> g = gt.random_graph(100, lambda: 2, directed=False)
>>> comp, art, nc = gt.label_biconnected_components(g)
>>> print comp.a
[0 0 3 2 0 3 1 0 2 0 0 0 1 3 2 0 1 0 1 0 1 1 2 1 0 3 1 0 1 0 0 0 2 0 0 0 1
 0 0 0 0 1 3 0 0 2 2 0 2 0 0 1 3 2 0 1 3 2 2 1 0 0 0 0 1 0 2 0 2 0 1 0 2 2
 0 2 0 2 1 0 1 0 0 0 1 0 2 0 1 1 0 1 0 1 2 0 0 2 2 2]
>>> print art.a
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
>>> print nc
4

graph_tool.topology.shortest_distance(g, source=None, weights=None, max_dist=None, directed=None, dense=False, dist_map=None, pred_map=False)

Calculate the distance of all vertices from a given source, or the all pairs shortest paths, if the source is not specified.

Parameters:

g : Graph

Graph to be used.

source : Vertex (optional, default: None)

Source vertex of the search. If unspecified, the all pairs shortest distances are computed.

weights : PropertyMap (optional, default: None)

The edge weights. If provided, the minimum spanning tree will minimize the edge weights.

max_dist : scalar value (optional, default: None)

If specified, this limits the maximum distance of the vertices are searched. This parameter has no effect if source == None.

directed : bool (optional, default:None)

Treat graph as directed or not, independently of its actual directionality.

dense : bool (optional, default: False)

If true, and source == None, the Floyd-Warshall algorithm is used, otherwise the Johnson algorithm is used. If source != None, this option has no effect.

dist_map : PropertyMap (optional, default: None)

Vertex property to store the distances. If none is supplied, one is created.

pred_map : bool (optional, default: False)

If true, a vertex property map with the predecessors is returned. Ignored if source=None.

Returns:

dist_map : PropertyMap

Vertex property map with the distances from source. If source is ‘None’, it will have a vector value type, with the distances to every vertex.

Notes

If a source is given, the distances are calculated with a breadth-first search (BFS) or Dijkstra’s algorithm [dijkstra], if weights are given. If source is not given, the distances are calculated with Johnson’s algorithm [johnson-apsp]. If dense=True, the Floyd-Warshall algorithm [floyd-warshall-apsp] is used instead.

If source is specified, the algorithm runs in O(V + E) time, or O(V \log V) if weights are given. If source is not specified, it runs in O(VE\log V) time, or O(V^3) if dense == True.

References

[bfs]

Edward Moore, “The shortest path through a maze”, International Symposium on the Theory of Switching (1959), Harvard University Press;http://www.boost.org/libs/graph/doc/breadth_first_search.html

[dijkstra]

E. Dijkstra, “A note on two problems in connexion with graphs.” Numerische Mathematik, 1:269-271, 1959. http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html

[johnson-apsp]

http://www.boost.org/libs/graph/doc/johnson_all_pairs_shortest.html

[floyd-warshall-apsp]

http://www.boost.org/libs/graph/doc/floyd_warshall_shortest.html

Examples

>>> from numpy.random import seed, poisson
>>> seed(42)
>>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
>>> dist = gt.shortest_distance(g, source=g.vertex(0))
>>> print dist.get_array()
[         0          4          4          4 2147483647          3
          3          2          2          3          4          4
          5          2          3          1          4          2
          4          4          6          2 2147483647          5
          2          4          4          5          4          5
          3          3          2          2          1          4
          2          4          5          4 2147483647          4
          3          3          3          3          3          3
          6          4          4          1          5          3
          3          2          2          5          2          4
          3          4          5          4          3 2147483647
          5          4          3          2          2 2147483647
          4          4          5          2          3          5
          4          5          4          4          3 2147483647
          5          3 2147483647          3          3          4
          3          3          3          1          5          5
          2          3          4          3]
>>> dist = gt.shortest_distance(g)
>>> print array(dist[g.vertex(0)])
[         0          4          4          4 2147483647          3
          3          2          2          3          4          4
          5          2          3          1          4          2
          4          4          6          2 2147483647          5
          2          4          4          5          4          5
          3          3          2          2          1          4
          2          4          5          4 2147483647          4
          3          3          3          3          3          3
          6          4          4          1          5          3
          3          2          2          5          2          4
          3          4          5          4          3 2147483647
          5          4          3          2          2 2147483647
          4          4          5          2          3          5
          4          5          4          4          3 2147483647
          5          3 2147483647          3          3          4
          3          3          3          1          5          5
          2          3          4          3]

graph_tool.topology.shortest_path(g, source, target, weights=None, pred_map=None)

Return the shortest path from source to target.

Parameters:

g : Graph

Graph to be used.

source : Vertex

Source vertex of the search.

source : Vertex

Target vertex of the search.

weights : PropertyMap (optional, default: None)

The edge weights. If provided, the minimum spanning tree will minimize the edge weights.

pred_map : PropertyMap (optional, default: None)

Vertex property map with the predecessors in the search tree. If this is provided, the shortest paths are not computed, and are obtained directly from this map.

Returns:

vertex_list : list of Vertex

List of vertices from source to target in the shortest path.

edge_list : list of Edge

List of edges from source to target in the shortest path.

Notes

The paths are computed with a breadth-first search (BFS) or Dijkstra’s algorithm [dijkstra], if weights are given.

The algorithm runs in O(V + E) time, or O(V \log V) if weights are given.

References

[bfs]

Edward Moore, “The shortest path through a maze”, International Symposium on the Theory of Switching (1959), Harvard University Press;http://www.boost.org/libs/graph/doc/breadth_first_search.html

[dijkstra]

E. Dijkstra, “A note on two problems in connexion with graphs.” Numerische Mathematik, 1:269-271, 1959. http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html

Examples

>>> from numpy.random import seed, poisson
>>> seed(42)
>>> g = gt.random_graph(300, lambda: (poisson(3), poisson(3)))
>>> vlist, elist = gt.shortest_path(g, g.vertex(10), g.vertex(11))
>>> print [str(v) for v in vlist]
['10', '283', '20', '158', '104', '25', '275', '75', '11']
>>> print [str(e) for e in elist]
['(10,283)', '(283,20)', '(20,158)', '(158,104)', '(104,25)', '(25,275)', '(275,75)', '(75,11)']

graph_tool.topology.is_planar(g, embedding=False, kuratowski=False)

Test if the graph is planar.

Parameters:

g : Graph

Graph to be used.

embedding : bool (optional, default: False)

If true, return a mapping from vertices to the clockwise order of out-edges in the planar embedding.

kuratowski : bool (optional, default: False)

If true, the minimal set of edges that form the obstructing Kuratowski subgraph will be returned as a property map, if the graph is not planar.

Returns:

is_planar : bool

Whether or not the graph is planar.

embedding : PropertyMap (only if embedding=True)

A vertex property map with the out-edges indexes in clockwise order in the planar embedding,

kuratowski : PropertyMap (only if kuratowski=True)

An edge property map with the minimal set of edges that form the obstructing Kuratowski subgraph (if the value of kuratowski[e] is 1, the edge belongs to the set)

Notes

A graph is planar if it can be drawn in two-dimensional space without any of its edges crossing. This algorithm performs the Boyer-Myrvold planarity testing [boyer-myrvold]. See [boost-planarity] for more details.

This algorithm runs in O(V) time.

References

[boyer-myrvold]

John M. Boyer and Wendy J. Myrvold, “On the Cutting Edge: Simplified O(n) Planarity by Edge Addition Journal of Graph Algorithms and Applications”, 8(2): 241-273, 2004.

[boost-planarity]

http://www.boost.org/libs/graph/doc/boyer_myrvold.html

Examples

>>> from numpy.random import seed, random
>>> seed(42)
>>> g = gt.triangulation(random((100,2)))[0]
>>> p, embed_order = gt.is_planar(g, embedding=True)
>>> print p
True
>>> print list(embed_order[g.vertex(0)])
[0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]
>>> g = gt.random_graph(100, lambda: 4, directed=False)
>>> p, kur = gt.is_planar(g, kuratowski=True)
>>> print p
False
>>> g.set_edge_filter(kur, True)
>>> gt.graph_draw(g, layout="arf",  size=(7,7), output="kuratowski.png")
<...>

Obstructing Kuratowski subgraph of a random graph.