from collections import defaultdict
import networkx as nx
__all__ = ["check_planarity", "PlanarEmbedding"]
[docs]def check_planarity(G, counterexample=False):
"""Check if a graph is planar and return a counterexample or an embedding.
A graph is planar iff it can be drawn in a plane without
any edge intersections.
Parameters
----------
G : NetworkX graph
counterexample : bool
A Kuratowski subgraph (to proof non planarity) is only returned if set
to true.
Returns
-------
(is_planar, certificate) : (bool, NetworkX graph) tuple
is_planar is true if the graph is planar.
If the graph is planar `certificate` is a PlanarEmbedding
otherwise it is a Kuratowski subgraph.
Notes
-----
A (combinatorial) embedding consists of cyclic orderings of the incident
edges at each vertex. Given such an embedding there are multiple approaches
discussed in literature to drawing the graph (subject to various
constraints, e.g. integer coordinates), see e.g. [2].
The planarity check algorithm and extraction of the combinatorial embedding
is based on the Left-Right Planarity Test [1].
A counterexample is only generated if the corresponding parameter is set,
because the complexity of the counterexample generation is higher.
References
----------
.. [1] Ulrik Brandes:
The Left-Right Planarity Test
2009
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.217.9208
.. [2] Takao Nishizeki, Md Saidur Rahman:
Planar graph drawing
Lecture Notes Series on Computing: Volume 12
2004
"""
planarity_state = LRPlanarity(G)
embedding = planarity_state.lr_planarity()
if embedding is None:
# graph is not planar
if counterexample:
return False, get_counterexample(G)
else:
return False, None
else:
# graph is planar
return True, embedding
def check_planarity_recursive(G, counterexample=False):
"""Recursive version of :meth:`check_planarity`."""
planarity_state = LRPlanarity(G)
embedding = planarity_state.lr_planarity_recursive()
if embedding is None:
# graph is not planar
if counterexample:
return False, get_counterexample_recursive(G)
else:
return False, None
else:
# graph is planar
return True, embedding
def get_counterexample(G):
"""Obtains a Kuratowski subgraph.
Raises nx.NetworkXException if G is planar.
The function removes edges such that the graph is still not planar.
At some point the removal of any edge would make the graph planar.
This subgraph must be a Kuratowski subgraph.
Parameters
----------
G : NetworkX graph
Returns
-------
subgraph : NetworkX graph
A Kuratowski subgraph that proves that G is not planar.
"""
# copy graph
G = nx.Graph(G)
if check_planarity(G)[0]:
raise nx.NetworkXException("G is planar - no counter example.")
# find Kuratowski subgraph
subgraph = nx.Graph()
for u in G:
nbrs = list(G[u])
for v in nbrs:
G.remove_edge(u, v)
if check_planarity(G)[0]:
G.add_edge(u, v)
subgraph.add_edge(u, v)
return subgraph
def get_counterexample_recursive(G):
"""Recursive version of :meth:`get_counterexample`.
"""
# copy graph
G = nx.Graph(G)
if check_planarity_recursive(G)[0]:
raise nx.NetworkXException("G is planar - no counter example.")
# find Kuratowski subgraph
subgraph = nx.Graph()
for u in G:
nbrs = list(G[u])
for v in nbrs:
G.remove_edge(u, v)
if check_planarity_recursive(G)[0]:
G.add_edge(u, v)
subgraph.add_edge(u, v)
return subgraph
class Interval:
"""Represents a set of return edges.
All return edges in an interval induce a same constraint on the contained
edges, which means that all edges must either have a left orientation or
all edges must have a right orientation.
"""
def __init__(self, low=None, high=None):
self.low = low
self.high = high
def empty(self):
"""Check if the interval is empty"""
return self.low is None and self.high is None
def copy(self):
"""Returns a copy of this interval"""
return Interval(self.low, self.high)
def conflicting(self, b, planarity_state):
"""Returns True if interval I conflicts with edge b"""
return (
not self.empty()
and planarity_state.lowpt[self.high] > planarity_state.lowpt[b]
)
class ConflictPair:
"""Represents a different constraint between two intervals.
The edges in the left interval must have a different orientation than
the one in the right interval.
"""
def __init__(self, left=Interval(), right=Interval()):
self.left = left
self.right = right
def swap(self):
"""Swap left and right intervals"""
temp = self.left
self.left = self.right
self.right = temp
def lowest(self, planarity_state):
"""Returns the lowest lowpoint of a conflict pair"""
if self.left.empty():
return planarity_state.lowpt[self.right.low]
if self.right.empty():
return planarity_state.lowpt[self.left.low]
return min(
planarity_state.lowpt[self.left.low], planarity_state.lowpt[self.right.low]
)
def top_of_stack(l):
"""Returns the element on top of the stack."""
if not l:
return None
return l[-1]
class LRPlanarity:
"""A class to maintain the state during planarity check."""
__slots__ = [
"G",
"roots",
"height",
"lowpt",
"lowpt2",
"nesting_depth",
"parent_edge",
"DG",
"adjs",
"ordered_adjs",
"ref",
"side",
"S",
"stack_bottom",
"lowpt_edge",
"left_ref",
"right_ref",
"embedding",
]
def __init__(self, G):
# copy G without adding self-loops
self.G = nx.Graph()
self.G.add_nodes_from(G.nodes)
for e in G.edges:
if e[0] != e[1]:
self.G.add_edge(e[0], e[1])
self.roots = []
# distance from tree root
self.height = defaultdict(lambda: None)
self.lowpt = {} # height of lowest return point of an edge
self.lowpt2 = {} # height of second lowest return point
self.nesting_depth = {} # for nesting order
# None -> missing edge
self.parent_edge = defaultdict(lambda: None)
# oriented DFS graph
self.DG = nx.DiGraph()
self.DG.add_nodes_from(G.nodes)
self.adjs = {}
self.ordered_adjs = {}
self.ref = defaultdict(lambda: None)
self.side = defaultdict(lambda: 1)
# stack of conflict pairs
self.S = []
self.stack_bottom = {}
self.lowpt_edge = {}
self.left_ref = {}
self.right_ref = {}
self.embedding = PlanarEmbedding()
def lr_planarity(self):
"""Execute the LR planarity test.
Returns
-------
embedding : dict
If the graph is planar an embedding is returned. Otherwise None.
"""
if self.G.order() > 2 and self.G.size() > 3 * self.G.order() - 6:
# graph is not planar
return None
# make adjacency lists for dfs
for v in self.G:
self.adjs[v] = list(self.G[v])
# orientation of the graph by depth first search traversal
for v in self.G:
if self.height[v] is None:
self.height[v] = 0
self.roots.append(v)
self.dfs_orientation(v)
# Free no longer used variables
self.G = None
self.lowpt2 = None
self.adjs = None
# testing
for v in self.DG: # sort the adjacency lists by nesting depth
# note: this sorting leads to non linear time
self.ordered_adjs[v] = sorted(
self.DG[v], key=lambda x: self.nesting_depth[(v, x)]
)
for v in self.roots:
if not self.dfs_testing(v):
return None
# Free no longer used variables
self.height = None
self.lowpt = None
self.S = None
self.stack_bottom = None
self.lowpt_edge = None
for e in self.DG.edges:
self.nesting_depth[e] = self.sign(e) * self.nesting_depth[e]
self.embedding.add_nodes_from(self.DG.nodes)
for v in self.DG:
# sort the adjacency lists again
self.ordered_adjs[v] = sorted(
self.DG[v], key=lambda x: self.nesting_depth[(v, x)]
)
# initialize the embedding
previous_node = None
for w in self.ordered_adjs[v]:
self.embedding.add_half_edge_cw(v, w, previous_node)
previous_node = w
# Free no longer used variables
self.DG = None
self.nesting_depth = None
self.ref = None
# compute the complete embedding
for v in self.roots:
self.dfs_embedding(v)
# Free no longer used variables
self.roots = None
self.parent_edge = None
self.ordered_adjs = None
self.left_ref = None
self.right_ref = None
self.side = None
return self.embedding
def lr_planarity_recursive(self):
"""Recursive version of :meth:`lr_planarity`."""
if self.G.order() > 2 and self.G.size() > 3 * self.G.order() - 6:
# graph is not planar
return None
# orientation of the graph by depth first search traversal
for v in self.G:
if self.height[v] is None:
self.height[v] = 0
self.roots.append(v)
self.dfs_orientation_recursive(v)
# Free no longer used variable
self.G = None
# testing
for v in self.DG: # sort the adjacency lists by nesting depth
# note: this sorting leads to non linear time
self.ordered_adjs[v] = sorted(
self.DG[v], key=lambda x: self.nesting_depth[(v, x)]
)
for v in self.roots:
if not self.dfs_testing_recursive(v):
return None
for e in self.DG.edges:
self.nesting_depth[e] = self.sign_recursive(e) * self.nesting_depth[e]
self.embedding.add_nodes_from(self.DG.nodes)
for v in self.DG:
# sort the adjacency lists again
self.ordered_adjs[v] = sorted(
self.DG[v], key=lambda x: self.nesting_depth[(v, x)]
)
# initialize the embedding
previous_node = None
for w in self.ordered_adjs[v]:
self.embedding.add_half_edge_cw(v, w, previous_node)
previous_node = w
# compute the complete embedding
for v in self.roots:
self.dfs_embedding_recursive(v)
return self.embedding
def dfs_orientation(self, v):
"""Orient the graph by DFS, compute lowpoints and nesting order.
"""
# the recursion stack
dfs_stack = [v]
# index of next edge to handle in adjacency list of each node
ind = defaultdict(lambda: 0)
# boolean to indicate whether to skip the initial work for an edge
skip_init = defaultdict(lambda: False)
while dfs_stack:
v = dfs_stack.pop()
e = self.parent_edge[v]
for w in self.adjs[v][ind[v] :]:
vw = (v, w)
if not skip_init[vw]:
if (v, w) in self.DG.edges or (w, v) in self.DG.edges:
ind[v] += 1
continue # the edge was already oriented
self.DG.add_edge(v, w) # orient the edge
self.lowpt[vw] = self.height[v]
self.lowpt2[vw] = self.height[v]
if self.height[w] is None: # (v, w) is a tree edge
self.parent_edge[w] = vw
self.height[w] = self.height[v] + 1
dfs_stack.append(v) # revisit v after finishing w
dfs_stack.append(w) # visit w next
skip_init[vw] = True # don't redo this block
break # handle next node in dfs_stack (i.e. w)
else: # (v, w) is a back edge
self.lowpt[vw] = self.height[w]
# determine nesting graph
self.nesting_depth[vw] = 2 * self.lowpt[vw]
if self.lowpt2[vw] < self.height[v]: # chordal
self.nesting_depth[vw] += 1
# update lowpoints of parent edge e
if e is not None:
if self.lowpt[vw] < self.lowpt[e]:
self.lowpt2[e] = min(self.lowpt[e], self.lowpt2[vw])
self.lowpt[e] = self.lowpt[vw]
elif self.lowpt[vw] > self.lowpt[e]:
self.lowpt2[e] = min(self.lowpt2[e], self.lowpt[vw])
else:
self.lowpt2[e] = min(self.lowpt2[e], self.lowpt2[vw])
ind[v] += 1
def dfs_orientation_recursive(self, v):
"""Recursive version of :meth:`dfs_orientation`."""
e = self.parent_edge[v]
for w in self.G[v]:
if (v, w) in self.DG.edges or (w, v) in self.DG.edges:
continue # the edge was already oriented
vw = (v, w)
self.DG.add_edge(v, w) # orient the edge
self.lowpt[vw] = self.height[v]
self.lowpt2[vw] = self.height[v]
if self.height[w] is None: # (v, w) is a tree edge
self.parent_edge[w] = vw
self.height[w] = self.height[v] + 1
self.dfs_orientation_recursive(w)
else: # (v, w) is a back edge
self.lowpt[vw] = self.height[w]
# determine nesting graph
self.nesting_depth[vw] = 2 * self.lowpt[vw]
if self.lowpt2[vw] < self.height[v]: # chordal
self.nesting_depth[vw] += 1
# update lowpoints of parent edge e
if e is not None:
if self.lowpt[vw] < self.lowpt[e]:
self.lowpt2[e] = min(self.lowpt[e], self.lowpt2[vw])
self.lowpt[e] = self.lowpt[vw]
elif self.lowpt[vw] > self.lowpt[e]:
self.lowpt2[e] = min(self.lowpt2[e], self.lowpt[vw])
else:
self.lowpt2[e] = min(self.lowpt2[e], self.lowpt2[vw])
def dfs_testing(self, v):
"""Test for LR partition."""
# the recursion stack
dfs_stack = [v]
# index of next edge to handle in adjacency list of each node
ind = defaultdict(lambda: 0)
# boolean to indicate whether to skip the initial work for an edge
skip_init = defaultdict(lambda: False)
while dfs_stack:
v = dfs_stack.pop()
e = self.parent_edge[v]
# to indicate whether to skip the final block after the for loop
skip_final = False
for w in self.ordered_adjs[v][ind[v] :]:
ei = (v, w)
if not skip_init[ei]:
self.stack_bottom[ei] = top_of_stack(self.S)
if ei == self.parent_edge[w]: # tree edge
dfs_stack.append(v) # revisit v after finishing w
dfs_stack.append(w) # visit w next
skip_init[ei] = True # don't redo this block
skip_final = True # skip final work after breaking
break # handle next node in dfs_stack (i.e. w)
else: # back edge
self.lowpt_edge[ei] = ei
self.S.append(ConflictPair(right=Interval(ei, ei)))
# integrate new return edges
if self.lowpt[ei] < self.height[v]:
if w == self.ordered_adjs[v][0]: # e_i has return edge
self.lowpt_edge[e] = self.lowpt_edge[ei]
else: # add constraints of e_i
if not self.add_constraints(ei, e):
# graph is not planar
return False
ind[v] += 1
if not skip_final:
# remove back edges returning to parent
if e is not None: # v isn't root
self.remove_back_edges(e)
return True
def dfs_testing_recursive(self, v):
"""Recursive version of :meth:`dfs_testing`."""
e = self.parent_edge[v]
for w in self.ordered_adjs[v]:
ei = (v, w)
self.stack_bottom[ei] = top_of_stack(self.S)
if ei == self.parent_edge[w]: # tree edge
if not self.dfs_testing_recursive(w):
return False
else: # back edge
self.lowpt_edge[ei] = ei
self.S.append(ConflictPair(right=Interval(ei, ei)))
# integrate new return edges
if self.lowpt[ei] < self.height[v]:
if w == self.ordered_adjs[v][0]: # e_i has return edge
self.lowpt_edge[e] = self.lowpt_edge[ei]
else: # add constraints of e_i
if not self.add_constraints(ei, e):
# graph is not planar
return False
# remove back edges returning to parent
if e is not None: # v isn't root
self.remove_back_edges(e)
return True
def add_constraints(self, ei, e):
P = ConflictPair()
# merge return edges of e_i into P.right
while True:
Q = self.S.pop()
if not Q.left.empty():
Q.swap()
if not Q.left.empty(): # not planar
return False
if self.lowpt[Q.right.low] > self.lowpt[e]:
# merge intervals
if P.right.empty(): # topmost interval
P.right = Q.right.copy()
else:
self.ref[P.right.low] = Q.right.high
P.right.low = Q.right.low
else: # align
self.ref[Q.right.low] = self.lowpt_edge[e]
if top_of_stack(self.S) == self.stack_bottom[ei]:
break
# merge conflicting return edges of e_1,...,e_i-1 into P.L
while top_of_stack(self.S).left.conflicting(ei, self) or top_of_stack(
self.S
).right.conflicting(ei, self):
Q = self.S.pop()
if Q.right.conflicting(ei, self):
Q.swap()
if Q.right.conflicting(ei, self): # not planar
return False
# merge interval below lowpt(e_i) into P.R
self.ref[P.right.low] = Q.right.high
if Q.right.low is not None:
P.right.low = Q.right.low
if P.left.empty(): # topmost interval
P.left = Q.left.copy()
else:
self.ref[P.left.low] = Q.left.high
P.left.low = Q.left.low
if not (P.left.empty() and P.right.empty()):
self.S.append(P)
return True
def remove_back_edges(self, e):
u = e[0]
# trim back edges ending at parent u
# drop entire conflict pairs
while self.S and top_of_stack(self.S).lowest(self) == self.height[u]:
P = self.S.pop()
if P.left.low is not None:
self.side[P.left.low] = -1
if self.S: # one more conflict pair to consider
P = self.S.pop()
# trim left interval
while P.left.high is not None and P.left.high[1] == u:
P.left.high = self.ref[P.left.high]
if P.left.high is None and P.left.low is not None:
# just emptied
self.ref[P.left.low] = P.right.low
self.side[P.left.low] = -1
P.left.low = None
# trim right interval
while P.right.high is not None and P.right.high[1] == u:
P.right.high = self.ref[P.right.high]
if P.right.high is None and P.right.low is not None:
# just emptied
self.ref[P.right.low] = P.left.low
self.side[P.right.low] = -1
P.right.low = None
self.S.append(P)
# side of e is side of a highest return edge
if self.lowpt[e] < self.height[u]: # e has return edge
hl = top_of_stack(self.S).left.high
hr = top_of_stack(self.S).right.high
if hl is not None and (hr is None or self.lowpt[hl] > self.lowpt[hr]):
self.ref[e] = hl
else:
self.ref[e] = hr
def dfs_embedding(self, v):
"""Completes the embedding."""
# the recursion stack
dfs_stack = [v]
# index of next edge to handle in adjacency list of each node
ind = defaultdict(lambda: 0)
while dfs_stack:
v = dfs_stack.pop()
for w in self.ordered_adjs[v][ind[v] :]:
ind[v] += 1
ei = (v, w)
if ei == self.parent_edge[w]: # tree edge
self.embedding.add_half_edge_first(w, v)
self.left_ref[v] = w
self.right_ref[v] = w
dfs_stack.append(v) # revisit v after finishing w
dfs_stack.append(w) # visit w next
break # handle next node in dfs_stack (i.e. w)
else: # back edge
if self.side[ei] == 1:
self.embedding.add_half_edge_cw(w, v, self.right_ref[w])
else:
self.embedding.add_half_edge_ccw(w, v, self.left_ref[w])
self.left_ref[w] = v
def dfs_embedding_recursive(self, v):
"""Recursive version of :meth:`dfs_embedding`."""
for w in self.ordered_adjs[v]:
ei = (v, w)
if ei == self.parent_edge[w]: # tree edge
self.embedding.add_half_edge_first(w, v)
self.left_ref[v] = w
self.right_ref[v] = w
self.dfs_embedding_recursive(w)
else: # back edge
if self.side[ei] == 1:
# place v directly after right_ref[w] in embed. list of w
self.embedding.add_half_edge_cw(w, v, self.right_ref[w])
else:
# place v directly before left_ref[w] in embed. list of w
self.embedding.add_half_edge_ccw(w, v, self.left_ref[w])
self.left_ref[w] = v
def sign(self, e):
"""Resolve the relative side of an edge to the absolute side."""
# the recursion stack
dfs_stack = [e]
# dict to remember reference edges
old_ref = defaultdict(lambda: None)
while dfs_stack:
e = dfs_stack.pop()
if self.ref[e] is not None:
dfs_stack.append(e) # revisit e after finishing self.ref[e]
dfs_stack.append(self.ref[e]) # visit self.ref[e] next
old_ref[e] = self.ref[e] # remember value of self.ref[e]
self.ref[e] = None
else:
self.side[e] *= self.side[old_ref[e]]
return self.side[e]
def sign_recursive(self, e):
"""Recursive version of :meth:`sign`."""
if self.ref[e] is not None:
self.side[e] = self.side[e] * self.sign_recursive(self.ref[e])
self.ref[e] = None
return self.side[e]
[docs]class PlanarEmbedding(nx.DiGraph):
"""Represents a planar graph with its planar embedding.
The planar embedding is given by a `combinatorial embedding
<https://en.wikipedia.org/wiki/Graph_embedding#Combinatorial_embedding>`_.
**Neighbor ordering:**
In comparison to a usual graph structure, the embedding also stores the
order of all neighbors for every vertex.
The order of the neighbors can be given in clockwise (cw) direction or
counterclockwise (ccw) direction. This order is stored as edge attributes
in the underlying directed graph. For the edge (u, v) the edge attribute
'cw' is set to the neighbor of u that follows immediately after v in
clockwise direction.
In order for a PlanarEmbedding to be valid it must fulfill multiple
conditions. It is possible to check if these conditions are fulfilled with
the method :meth:`check_structure`.
The conditions are:
* Edges must go in both directions (because the edge attributes differ)
* Every edge must have a 'cw' and 'ccw' attribute which corresponds to a
correct planar embedding.
* A node with non zero degree must have a node attribute 'first_nbr'.
As long as a PlanarEmbedding is invalid only the following methods should
be called:
* :meth:`add_half_edge_ccw`
* :meth:`add_half_edge_cw`
* :meth:`connect_components`
* :meth:`add_half_edge_first`
Even though the graph is a subclass of nx.DiGraph, it can still be used
for algorithms that require undirected graphs, because the method
:meth:`is_directed` is overridden. This is possible, because a valid
PlanarGraph must have edges in both directions.
**Half edges:**
In methods like `add_half_edge_ccw` the term "half-edge" is used, which is
a term that is used in `doubly connected edge lists
<https://en.wikipedia.org/wiki/Doubly_connected_edge_list>`_. It is used
to emphasize that the edge is only in one direction and there exists
another half-edge in the opposite direction.
While conventional edges always have two faces (including outer face) next
to them, it is possible to assign each half-edge *exactly one* face.
For a half-edge (u, v) that is orientated such that u is below v then the
face that belongs to (u, v) is to the right of this half-edge.
Examples
--------
Create an embedding of a star graph (compare `nx.star_graph(3)`):
>>> G = nx.PlanarEmbedding()
>>> G.add_half_edge_cw(0, 1, None)
>>> G.add_half_edge_cw(0, 2, 1)
>>> G.add_half_edge_cw(0, 3, 2)
>>> G.add_half_edge_cw(1, 0, None)
>>> G.add_half_edge_cw(2, 0, None)
>>> G.add_half_edge_cw(3, 0, None)
Alternatively the same embedding can also be defined in counterclockwise
orientation. The following results in exactly the same PlanarEmbedding:
>>> G = nx.PlanarEmbedding()
>>> G.add_half_edge_ccw(0, 1, None)
>>> G.add_half_edge_ccw(0, 3, 1)
>>> G.add_half_edge_ccw(0, 2, 3)
>>> G.add_half_edge_ccw(1, 0, None)
>>> G.add_half_edge_ccw(2, 0, None)
>>> G.add_half_edge_ccw(3, 0, None)
After creating a graph, it is possible to validate that the PlanarEmbedding
object is correct:
>>> G.check_structure()
"""
[docs] def get_data(self):
"""Converts the adjacency structure into a better readable structure.
Returns
-------
embedding : dict
A dict mapping all nodes to a list of neighbors sorted in
clockwise order.
See Also
--------
set_data
"""
embedding = dict()
for v in self:
embedding[v] = list(self.neighbors_cw_order(v))
return embedding
[docs] def set_data(self, data):
"""Inserts edges according to given sorted neighbor list.
The input format is the same as the output format of get_data().
Parameters
----------
data : dict
A dict mapping all nodes to a list of neighbors sorted in
clockwise order.
See Also
--------
get_data
"""
for v in data:
for w in reversed(data[v]):
self.add_half_edge_first(v, w)
[docs] def neighbors_cw_order(self, v):
"""Generator for the neighbors of v in clockwise order.
Parameters
----------
v : node
Yields
------
node
"""
if len(self[v]) == 0:
# v has no neighbors
return
start_node = self.nodes[v]["first_nbr"]
yield start_node
current_node = self[v][start_node]["cw"]
while start_node != current_node:
yield current_node
current_node = self[v][current_node]["cw"]
[docs] def check_structure(self):
"""Runs without exceptions if this object is valid.
Checks that the following properties are fulfilled:
* Edges go in both directions (because the edge attributes differ).
* Every edge has a 'cw' and 'ccw' attribute which corresponds to a
correct planar embedding.
* A node with a degree larger than 0 has a node attribute 'first_nbr'.
Running this method verifies that the underlying Graph must be planar.
Raises
------
NetworkXException
This exception is raised with a short explanation if the
PlanarEmbedding is invalid.
"""
# Check fundamental structure
for v in self:
try:
sorted_nbrs = set(self.neighbors_cw_order(v))
except KeyError as e:
msg = f"Bad embedding. Missing orientation for a neighbor of {v}"
raise nx.NetworkXException(msg) from e
unsorted_nbrs = set(self[v])
if sorted_nbrs != unsorted_nbrs:
msg = "Bad embedding. Edge orientations not set correctly."
raise nx.NetworkXException(msg)
for w in self[v]:
# Check if opposite half-edge exists
if not self.has_edge(w, v):
msg = "Bad embedding. Opposite half-edge is missing."
raise nx.NetworkXException(msg)
# Check planarity
counted_half_edges = set()
for component in nx.connected_components(self):
if len(component) == 1:
# Don't need to check single node component
continue
num_nodes = len(component)
num_half_edges = 0
num_faces = 0
for v in component:
for w in self.neighbors_cw_order(v):
num_half_edges += 1
if (v, w) not in counted_half_edges:
# We encountered a new face
num_faces += 1
# Mark all half-edges belonging to this face
self.traverse_face(v, w, counted_half_edges)
num_edges = num_half_edges // 2 # num_half_edges is even
if num_nodes - num_edges + num_faces != 2:
# The result does not match Euler's formula
msg = "Bad embedding. The graph does not match Euler's formula"
raise nx.NetworkXException(msg)
[docs] def add_half_edge_ccw(self, start_node, end_node, reference_neighbor):
"""Adds a half-edge from start_node to end_node.
The half-edge is added counter clockwise next to the existing half-edge
(start_node, reference_neighbor).
Parameters
----------
start_node : node
Start node of inserted edge.
end_node : node
End node of inserted edge.
reference_neighbor: node
End node of reference edge.
Raises
------
NetworkXException
If the reference_neighbor does not exist.
See Also
--------
add_half_edge_cw
connect_components
add_half_edge_first
"""
if reference_neighbor is None:
# The start node has no neighbors
self.add_edge(start_node, end_node) # Add edge to graph
self[start_node][end_node]["cw"] = end_node
self[start_node][end_node]["ccw"] = end_node
self.nodes[start_node]["first_nbr"] = end_node
else:
ccw_reference = self[start_node][reference_neighbor]["ccw"]
self.add_half_edge_cw(start_node, end_node, ccw_reference)
if reference_neighbor == self.nodes[start_node].get("first_nbr", None):
# Update first neighbor
self.nodes[start_node]["first_nbr"] = end_node
[docs] def add_half_edge_cw(self, start_node, end_node, reference_neighbor):
"""Adds a half-edge from start_node to end_node.
The half-edge is added clockwise next to the existing half-edge
(start_node, reference_neighbor).
Parameters
----------
start_node : node
Start node of inserted edge.
end_node : node
End node of inserted edge.
reference_neighbor: node
End node of reference edge.
Raises
------
NetworkXException
If the reference_neighbor does not exist.
See Also
--------
add_half_edge_ccw
connect_components
add_half_edge_first
"""
self.add_edge(start_node, end_node) # Add edge to graph
if reference_neighbor is None:
# The start node has no neighbors
self[start_node][end_node]["cw"] = end_node
self[start_node][end_node]["ccw"] = end_node
self.nodes[start_node]["first_nbr"] = end_node
return
if reference_neighbor not in self[start_node]:
raise nx.NetworkXException(
"Cannot add edge. Reference neighbor does not exist"
)
# Get half-edge at the other side
cw_reference = self[start_node][reference_neighbor]["cw"]
# Alter half-edge data structures
self[start_node][reference_neighbor]["cw"] = end_node
self[start_node][end_node]["cw"] = cw_reference
self[start_node][cw_reference]["ccw"] = end_node
self[start_node][end_node]["ccw"] = reference_neighbor
[docs] def connect_components(self, v, w):
"""Adds half-edges for (v, w) and (w, v) at some position.
This method should only be called if v and w are in different
components, or it might break the embedding.
This especially means that if `connect_components(v, w)`
is called it is not allowed to call `connect_components(w, v)`
afterwards. The neighbor orientations in both directions are
all set correctly after the first call.
Parameters
----------
v : node
w : node
See Also
--------
add_half_edge_ccw
add_half_edge_cw
add_half_edge_first
"""
self.add_half_edge_first(v, w)
self.add_half_edge_first(w, v)
[docs] def add_half_edge_first(self, start_node, end_node):
"""The added half-edge is inserted at the first position in the order.
Parameters
----------
start_node : node
end_node : node
See Also
--------
add_half_edge_ccw
add_half_edge_cw
connect_components
"""
if start_node in self and "first_nbr" in self.nodes[start_node]:
reference = self.nodes[start_node]["first_nbr"]
else:
reference = None
self.add_half_edge_ccw(start_node, end_node, reference)
[docs] def next_face_half_edge(self, v, w):
"""Returns the following half-edge left of a face.
Parameters
----------
v : node
w : node
Returns
-------
half-edge : tuple
"""
new_node = self[w][v]["ccw"]
return w, new_node
[docs] def traverse_face(self, v, w, mark_half_edges=None):
"""Returns nodes on the face that belong to the half-edge (v, w).
The face that is traversed lies to the right of the half-edge (in an
orientation where v is below w).
Optionally it is possible to pass a set to which all encountered half
edges are added. Before calling this method, this set must not include
any half-edges that belong to the face.
Parameters
----------
v : node
Start node of half-edge.
w : node
End node of half-edge.
mark_half_edges: set, optional
Set to which all encountered half-edges are added.
Returns
-------
face : list
A list of nodes that lie on this face.
"""
if mark_half_edges is None:
mark_half_edges = set()
face_nodes = [v]
mark_half_edges.add((v, w))
prev_node = v
cur_node = w
# Last half-edge is (incoming_node, v)
incoming_node = self[v][w]["cw"]
while cur_node != v or prev_node != incoming_node:
face_nodes.append(cur_node)
prev_node, cur_node = self.next_face_half_edge(prev_node, cur_node)
if (prev_node, cur_node) in mark_half_edges:
raise nx.NetworkXException("Bad planar embedding. Impossible face.")
mark_half_edges.add((prev_node, cur_node))
return face_nodes
[docs] def is_directed(self):
"""A valid PlanarEmbedding is undirected.
All reverse edges are contained, i.e. for every existing
half-edge (v, w) the half-edge in the opposite direction (w, v) is also
contained.
"""
return False