from collections import defaultdict
import networkx as nx
__all__ = ["check_planarity", "is_planar", "PlanarEmbedding"]
[docs]
@nx._dispatchable
def is_planar(G):
"""Returns True if and only if `G` is planar.
A graph is *planar* iff it can be drawn in a plane without
any edge intersections.
Parameters
----------
G : NetworkX graph
Returns
-------
bool
Whether the graph is planar.
Examples
--------
>>> G = nx.Graph([(0, 1), (0, 2)])
>>> nx.is_planar(G)
True
>>> nx.is_planar(nx.complete_graph(5))
False
See Also
--------
check_planarity :
Check if graph is planar *and* return a `PlanarEmbedding` instance if True.
"""
return check_planarity(G, counterexample=False)[0]
[docs]
@nx._dispatchable(returns_graph=True)
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.
Examples
--------
>>> G = nx.Graph([(0, 1), (0, 2)])
>>> is_planar, P = nx.check_planarity(G)
>>> print(is_planar)
True
When `G` is planar, a `PlanarEmbedding` instance is returned:
>>> P.get_data()
{0: [1, 2], 1: [0], 2: [0]}
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.
See also
--------
is_planar :
Check for planarity without creating a `PlanarEmbedding` or counterexample.
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
@nx._dispatchable(returns_graph=True)
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
@nx._dispatchable(returns_graph=True)
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
@nx._dispatchable(returns_graph=True)
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(v, w, ccw=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(v, w, ccw=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(w, v, ccw=self.right_ref[w])
else:
self.embedding.add_half_edge(w, v, cw=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(w, v, ccw=self.right_ref[w])
else:
# place v directly before left_ref[w] in embed. list of w
self.embedding.add_half_edge(w, v, cw=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>`_.
.. note:: `check_planarity` is the preferred way to check if a graph is planar.
**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.
As long as a PlanarEmbedding is invalid only the following methods should
be called:
* :meth:`add_half_edge`
* :meth:`connect_components`
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` 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 oriented such that u is below v then the
face that belongs to (u, v) is to the right of this half-edge.
See Also
--------
is_planar :
Preferred way to check if an existing graph is planar.
check_planarity :
A convenient way to create a `PlanarEmbedding`. If not planar,
it returns a subgraph that shows this.
Examples
--------
Create an embedding of a star graph (compare `nx.star_graph(3)`):
>>> G = nx.PlanarEmbedding()
>>> G.add_half_edge(0, 1)
>>> G.add_half_edge(0, 2, ccw=1)
>>> G.add_half_edge(0, 3, ccw=2)
>>> G.add_half_edge(1, 0)
>>> G.add_half_edge(2, 0)
>>> G.add_half_edge(3, 0)
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(0, 1)
>>> G.add_half_edge(0, 3, cw=1)
>>> G.add_half_edge(0, 2, cw=3)
>>> G.add_half_edge(1, 0)
>>> G.add_half_edge(2, 0)
>>> G.add_half_edge(3, 0)
After creating a graph, it is possible to validate that the PlanarEmbedding
object is correct:
>>> G.check_structure()
"""
[docs]
def __init__(self, incoming_graph_data=None, **attr):
super().__init__(incoming_graph_data=incoming_graph_data, **attr)
self.add_edge = self.__forbidden
self.add_edges_from = self.__forbidden
self.add_weighted_edges_from = self.__forbidden
def __forbidden(self, *args, **kwargs):
"""Forbidden operation
Any edge additions to a PlanarEmbedding should be done using
method `add_half_edge`.
"""
raise NotImplementedError(
"Use `add_half_edge` method to add edges to a PlanarEmbedding."
)
[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 = {}
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:
ref = None
for w in reversed(data[v]):
self.add_half_edge(v, w, cw=ref)
ref = w
[docs]
def remove_node(self, n):
"""Remove node n.
Removes the node n and all adjacent edges, updating the
PlanarEmbedding to account for any resulting edge removal.
Attempting to remove a non-existent node will raise an exception.
Parameters
----------
n : node
A node in the graph
Raises
------
NetworkXError
If n is not in the graph.
See Also
--------
remove_nodes_from
"""
try:
for u in self._pred[n]:
succs_u = self._succ[u]
un_cw = succs_u[n]["cw"]
un_ccw = succs_u[n]["ccw"]
del succs_u[n]
del self._pred[u][n]
if n != un_cw:
succs_u[un_cw]["ccw"] = un_ccw
succs_u[un_ccw]["cw"] = un_cw
del self._node[n]
del self._succ[n]
del self._pred[n]
except KeyError as err: # NetworkXError if n not in self
raise nx.NetworkXError(
f"The node {n} is not in the planar embedding."
) from err
nx._clear_cache(self)
[docs]
def remove_nodes_from(self, nodes):
"""Remove multiple nodes.
Parameters
----------
nodes : iterable container
A container of nodes (list, dict, set, etc.). If a node
in the container is not in the graph it is silently ignored.
See Also
--------
remove_node
Notes
-----
When removing nodes from an iterator over the graph you are changing,
a `RuntimeError` will be raised with message:
`RuntimeError: dictionary changed size during iteration`. This
happens when the graph's underlying dictionary is modified during
iteration. To avoid this error, evaluate the iterator into a separate
object, e.g. by using `list(iterator_of_nodes)`, and pass this
object to `G.remove_nodes_from`.
"""
for n in nodes:
if n in self._node:
self.remove_node(n)
# silently skip non-existing nodes
[docs]
def neighbors_cw_order(self, v):
"""Generator for the neighbors of v in clockwise order.
Parameters
----------
v : node
Yields
------
node
"""
succs = self._succ[v]
if not succs:
# v has no neighbors
return
start_node = next(reversed(succs))
yield start_node
current_node = succs[start_node]["cw"]
while start_node != current_node:
yield current_node
current_node = succs[current_node]["cw"]
[docs]
def add_half_edge(self, start_node, end_node, *, cw=None, ccw=None):
"""Adds a half-edge from `start_node` to `end_node`.
If the half-edge is not the first one out of `start_node`, a reference
node must be provided either in the clockwise (parameter `cw`) or in
the counterclockwise (parameter `ccw`) direction. Only one of `cw`/`ccw`
can be specified (or neither in the case of the first edge).
Note that specifying a reference in the clockwise (`cw`) direction means
inserting the new edge in the first counterclockwise position with
respect to the reference (and vice-versa).
Parameters
----------
start_node : node
Start node of inserted edge.
end_node : node
End node of inserted edge.
cw, ccw: node
End node of reference edge.
Omit or pass `None` if adding the first out-half-edge of `start_node`.
Raises
------
NetworkXException
If the `cw` or `ccw` node is not a successor of `start_node`.
If `start_node` has successors, but neither `cw` or `ccw` is provided.
If both `cw` and `ccw` are specified.
See Also
--------
connect_components
"""
succs = self._succ.get(start_node)
if succs:
# there is already some edge out of start_node
leftmost_nbr = next(reversed(self._succ[start_node]))
if cw is not None:
if cw not in succs:
raise nx.NetworkXError("Invalid clockwise reference node.")
if ccw is not None:
raise nx.NetworkXError("Only one of cw/ccw can be specified.")
ref_ccw = succs[cw]["ccw"]
super().add_edge(start_node, end_node, cw=cw, ccw=ref_ccw)
succs[ref_ccw]["cw"] = end_node
succs[cw]["ccw"] = end_node
# when (cw == leftmost_nbr), the newly added neighbor is
# already at the end of dict self._succ[start_node] and
# takes the place of the former leftmost_nbr
move_leftmost_nbr_to_end = cw != leftmost_nbr
elif ccw is not None:
if ccw not in succs:
raise nx.NetworkXError("Invalid counterclockwise reference node.")
ref_cw = succs[ccw]["cw"]
super().add_edge(start_node, end_node, cw=ref_cw, ccw=ccw)
succs[ref_cw]["ccw"] = end_node
succs[ccw]["cw"] = end_node
move_leftmost_nbr_to_end = True
else:
raise nx.NetworkXError(
"Node already has out-half-edge(s), either cw or ccw reference node required."
)
if move_leftmost_nbr_to_end:
# LRPlanarity (via self.add_half_edge_first()) requires that
# we keep track of the leftmost neighbor, which we accomplish
# by keeping it as the last key in dict self._succ[start_node]
succs[leftmost_nbr] = succs.pop(leftmost_nbr)
else:
if cw is not None or ccw is not None:
raise nx.NetworkXError("Invalid reference node.")
# adding the first edge out of start_node
super().add_edge(start_node, end_node, ccw=end_node, cw=end_node)
[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.
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 err:
msg = f"Bad embedding. Missing orientation for a neighbor of {v}"
raise nx.NetworkXException(msg) from err
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
add_half_edge_cw
connect_components
"""
self.add_half_edge(start_node, end_node, cw=reference_neighbor)
[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
add_half_edge_ccw
connect_components
"""
self.add_half_edge(start_node, end_node, ccw=reference_neighbor)
[docs]
def remove_edge(self, u, v):
"""Remove the edge between u and v.
Parameters
----------
u, v : nodes
Remove the half-edges (u, v) and (v, u) and update the
edge ordering around the removed edge.
Raises
------
NetworkXError
If there is not an edge between u and v.
See Also
--------
remove_edges_from : remove a collection of edges
"""
try:
succs_u = self._succ[u]
succs_v = self._succ[v]
uv_cw = succs_u[v]["cw"]
uv_ccw = succs_u[v]["ccw"]
vu_cw = succs_v[u]["cw"]
vu_ccw = succs_v[u]["ccw"]
del succs_u[v]
del self._pred[v][u]
del succs_v[u]
del self._pred[u][v]
if v != uv_cw:
succs_u[uv_cw]["ccw"] = uv_ccw
succs_u[uv_ccw]["cw"] = uv_cw
if u != vu_cw:
succs_v[vu_cw]["ccw"] = vu_ccw
succs_v[vu_ccw]["cw"] = vu_cw
except KeyError as err:
raise nx.NetworkXError(
f"The edge {u}-{v} is not in the planar embedding."
) from err
nx._clear_cache(self)
[docs]
def remove_edges_from(self, ebunch):
"""Remove all edges specified in ebunch.
Parameters
----------
ebunch: list or container of edge tuples
Each pair of half-edges between the nodes given in the tuples
will be removed from the graph. The nodes can be passed as:
- 2-tuples (u, v) half-edges (u, v) and (v, u).
- 3-tuples (u, v, k) where k is ignored.
See Also
--------
remove_edge : remove a single edge
Notes
-----
Will fail silently if an edge in ebunch is not in the graph.
Examples
--------
>>> G = nx.path_graph(4) # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> ebunch = [(1, 2), (2, 3)]
>>> G.remove_edges_from(ebunch)
"""
for e in ebunch:
u, v = e[:2] # ignore edge data
# assuming that the PlanarEmbedding is valid, if the half_edge
# (u, v) is in the graph, then so is half_edge (v, u)
if u in self._succ and v in self._succ[u]:
self.remove_edge(u, v)
[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
"""
if v in self._succ and self._succ[v]:
ref = next(reversed(self._succ[v]))
else:
ref = None
self.add_half_edge(v, w, cw=ref)
if w in self._succ and self._succ[w]:
ref = next(reversed(self._succ[w]))
else:
ref = None
self.add_half_edge(w, v, cw=ref)
[docs]
def add_half_edge_first(self, start_node, end_node):
"""Add a half-edge and set end_node as start_node's leftmost neighbor.
The new edge is inserted counterclockwise with respect to the current
leftmost neighbor, if there is one.
Parameters
----------
start_node : node
end_node : node
See Also
--------
add_half_edge
connect_components
"""
succs = self._succ.get(start_node)
# the leftmost neighbor is the last entry in the
# self._succ[start_node] dict
leftmost_nbr = next(reversed(succs)) if succs else None
self.add_half_edge(start_node, end_node, cw=leftmost_nbr)
[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
[docs]
def copy(self, as_view=False):
if as_view is True:
return nx.graphviews.generic_graph_view(self)
G = self.__class__()
G.graph.update(self.graph)
G.add_nodes_from((n, d.copy()) for n, d in self._node.items())
super(self.__class__, G).add_edges_from(
(u, v, datadict.copy())
for u, nbrs in self._adj.items()
for v, datadict in nbrs.items()
)
return G
