85 lines
3.1 KiB
Python
85 lines
3.1 KiB
Python
"""
|
|
Utilities for connectivity package
|
|
"""
|
|
import networkx as nx
|
|
|
|
__all__ = ["build_auxiliary_node_connectivity", "build_auxiliary_edge_connectivity"]
|
|
|
|
|
|
def build_auxiliary_node_connectivity(G):
|
|
r"""Creates a directed graph D from an undirected graph G to compute flow
|
|
based node connectivity.
|
|
|
|
For an undirected graph G having `n` nodes and `m` edges we derive a
|
|
directed graph D with `2n` nodes and `2m+n` arcs by replacing each
|
|
original node `v` with two nodes `vA`, `vB` linked by an (internal)
|
|
arc in D. Then for each edge (`u`, `v`) in G we add two arcs (`uB`, `vA`)
|
|
and (`vB`, `uA`) in D. Finally we set the attribute capacity = 1 for each
|
|
arc in D [1]_.
|
|
|
|
For a directed graph having `n` nodes and `m` arcs we derive a
|
|
directed graph D with `2n` nodes and `m+n` arcs by replacing each
|
|
original node `v` with two nodes `vA`, `vB` linked by an (internal)
|
|
arc (`vA`, `vB`) in D. Then for each arc (`u`, `v`) in G we add one
|
|
arc (`uB`, `vA`) in D. Finally we set the attribute capacity = 1 for
|
|
each arc in D.
|
|
|
|
A dictionary with a mapping between nodes in the original graph and the
|
|
auxiliary digraph is stored as a graph attribute: H.graph['mapping'].
|
|
|
|
References
|
|
----------
|
|
.. [1] Kammer, Frank and Hanjo Taubig. Graph Connectivity. in Brandes and
|
|
Erlebach, 'Network Analysis: Methodological Foundations', Lecture
|
|
Notes in Computer Science, Volume 3418, Springer-Verlag, 2005.
|
|
http://www.informatik.uni-augsburg.de/thi/personen/kammer/Graph_Connectivity.pdf
|
|
|
|
"""
|
|
directed = G.is_directed()
|
|
|
|
mapping = {}
|
|
H = nx.DiGraph()
|
|
|
|
for i, node in enumerate(G):
|
|
mapping[node] = i
|
|
H.add_node(f"{i}A", id=node)
|
|
H.add_node(f"{i}B", id=node)
|
|
H.add_edge(f"{i}A", f"{i}B", capacity=1)
|
|
|
|
edges = []
|
|
for (source, target) in G.edges():
|
|
edges.append((f"{mapping[source]}B", f"{mapping[target]}A"))
|
|
if not directed:
|
|
edges.append((f"{mapping[target]}B", f"{mapping[source]}A"))
|
|
H.add_edges_from(edges, capacity=1)
|
|
|
|
# Store mapping as graph attribute
|
|
H.graph["mapping"] = mapping
|
|
return H
|
|
|
|
|
|
def build_auxiliary_edge_connectivity(G):
|
|
"""Auxiliary digraph for computing flow based edge connectivity
|
|
|
|
If the input graph is undirected, we replace each edge (`u`,`v`) with
|
|
two reciprocal arcs (`u`, `v`) and (`v`, `u`) and then we set the attribute
|
|
'capacity' for each arc to 1. If the input graph is directed we simply
|
|
add the 'capacity' attribute. Part of algorithm 1 in [1]_ .
|
|
|
|
References
|
|
----------
|
|
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. (this is a
|
|
chapter, look for the reference of the book).
|
|
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
|
|
"""
|
|
if G.is_directed():
|
|
H = nx.DiGraph()
|
|
H.add_nodes_from(G.nodes())
|
|
H.add_edges_from(G.edges(), capacity=1)
|
|
return H
|
|
else:
|
|
H = nx.DiGraph()
|
|
H.add_nodes_from(G.nodes())
|
|
for (source, target) in G.edges():
|
|
H.add_edges_from([(source, target), (target, source)], capacity=1)
|
|
return H
|