"""
Utilities for connectivity package
"""
import networkx as nx
__all__ = ["build_auxiliary_node_connectivity", "build_auxiliary_edge_connectivity"]
[docs]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
[docs]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