1116 lines
37 KiB
Python
1116 lines
37 KiB
Python
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from collections import defaultdict
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import networkx as nx
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__all__ = ["check_planarity", "PlanarEmbedding"]
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def check_planarity(G, counterexample=False):
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"""Check if a graph is planar and return a counterexample or an embedding.
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A graph is planar iff it can be drawn in a plane without
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any edge intersections.
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Parameters
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----------
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G : NetworkX graph
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counterexample : bool
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A Kuratowski subgraph (to proof non planarity) is only returned if set
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to true.
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Returns
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-------
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(is_planar, certificate) : (bool, NetworkX graph) tuple
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is_planar is true if the graph is planar.
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If the graph is planar `certificate` is a PlanarEmbedding
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otherwise it is a Kuratowski subgraph.
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Notes
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-----
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A (combinatorial) embedding consists of cyclic orderings of the incident
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edges at each vertex. Given such an embedding there are multiple approaches
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discussed in literature to drawing the graph (subject to various
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constraints, e.g. integer coordinates), see e.g. [2].
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The planarity check algorithm and extraction of the combinatorial embedding
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is based on the Left-Right Planarity Test [1].
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A counterexample is only generated if the corresponding parameter is set,
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because the complexity of the counterexample generation is higher.
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References
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----------
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.. [1] Ulrik Brandes:
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The Left-Right Planarity Test
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2009
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.217.9208
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.. [2] Takao Nishizeki, Md Saidur Rahman:
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Planar graph drawing
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Lecture Notes Series on Computing: Volume 12
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2004
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"""
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planarity_state = LRPlanarity(G)
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embedding = planarity_state.lr_planarity()
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if embedding is None:
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# graph is not planar
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if counterexample:
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return False, get_counterexample(G)
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else:
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return False, None
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else:
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# graph is planar
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return True, embedding
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def check_planarity_recursive(G, counterexample=False):
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"""Recursive version of :meth:`check_planarity`."""
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planarity_state = LRPlanarity(G)
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embedding = planarity_state.lr_planarity_recursive()
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if embedding is None:
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# graph is not planar
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if counterexample:
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return False, get_counterexample_recursive(G)
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else:
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return False, None
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else:
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# graph is planar
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return True, embedding
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def get_counterexample(G):
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"""Obtains a Kuratowski subgraph.
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Raises nx.NetworkXException if G is planar.
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The function removes edges such that the graph is still not planar.
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At some point the removal of any edge would make the graph planar.
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This subgraph must be a Kuratowski subgraph.
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Parameters
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----------
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G : NetworkX graph
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Returns
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-------
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subgraph : NetworkX graph
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A Kuratowski subgraph that proves that G is not planar.
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"""
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# copy graph
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G = nx.Graph(G)
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if check_planarity(G)[0]:
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raise nx.NetworkXException("G is planar - no counter example.")
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# find Kuratowski subgraph
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subgraph = nx.Graph()
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for u in G:
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nbrs = list(G[u])
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for v in nbrs:
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G.remove_edge(u, v)
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if check_planarity(G)[0]:
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G.add_edge(u, v)
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subgraph.add_edge(u, v)
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return subgraph
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def get_counterexample_recursive(G):
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"""Recursive version of :meth:`get_counterexample`.
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"""
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# copy graph
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G = nx.Graph(G)
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if check_planarity_recursive(G)[0]:
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raise nx.NetworkXException("G is planar - no counter example.")
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# find Kuratowski subgraph
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subgraph = nx.Graph()
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for u in G:
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nbrs = list(G[u])
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for v in nbrs:
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G.remove_edge(u, v)
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if check_planarity_recursive(G)[0]:
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G.add_edge(u, v)
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subgraph.add_edge(u, v)
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return subgraph
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class Interval:
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"""Represents a set of return edges.
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All return edges in an interval induce a same constraint on the contained
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edges, which means that all edges must either have a left orientation or
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all edges must have a right orientation.
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"""
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def __init__(self, low=None, high=None):
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self.low = low
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self.high = high
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def empty(self):
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"""Check if the interval is empty"""
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return self.low is None and self.high is None
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def copy(self):
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"""Returns a copy of this interval"""
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return Interval(self.low, self.high)
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def conflicting(self, b, planarity_state):
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"""Returns True if interval I conflicts with edge b"""
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return (
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not self.empty()
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and planarity_state.lowpt[self.high] > planarity_state.lowpt[b]
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)
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class ConflictPair:
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"""Represents a different constraint between two intervals.
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The edges in the left interval must have a different orientation than
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the one in the right interval.
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"""
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def __init__(self, left=Interval(), right=Interval()):
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self.left = left
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self.right = right
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def swap(self):
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"""Swap left and right intervals"""
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temp = self.left
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self.left = self.right
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self.right = temp
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def lowest(self, planarity_state):
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"""Returns the lowest lowpoint of a conflict pair"""
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if self.left.empty():
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return planarity_state.lowpt[self.right.low]
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if self.right.empty():
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return planarity_state.lowpt[self.left.low]
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return min(
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planarity_state.lowpt[self.left.low], planarity_state.lowpt[self.right.low]
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)
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def top_of_stack(l):
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"""Returns the element on top of the stack."""
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if not l:
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return None
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return l[-1]
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class LRPlanarity:
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"""A class to maintain the state during planarity check."""
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__slots__ = [
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"G",
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"roots",
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"height",
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"lowpt",
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"lowpt2",
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"nesting_depth",
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"parent_edge",
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"DG",
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"adjs",
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"ordered_adjs",
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"ref",
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"side",
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"S",
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"stack_bottom",
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"lowpt_edge",
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"left_ref",
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"right_ref",
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"embedding",
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]
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def __init__(self, G):
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# copy G without adding self-loops
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self.G = nx.Graph()
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self.G.add_nodes_from(G.nodes)
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for e in G.edges:
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if e[0] != e[1]:
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self.G.add_edge(e[0], e[1])
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self.roots = []
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# distance from tree root
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self.height = defaultdict(lambda: None)
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self.lowpt = {} # height of lowest return point of an edge
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self.lowpt2 = {} # height of second lowest return point
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self.nesting_depth = {} # for nesting order
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# None -> missing edge
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self.parent_edge = defaultdict(lambda: None)
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# oriented DFS graph
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self.DG = nx.DiGraph()
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self.DG.add_nodes_from(G.nodes)
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self.adjs = {}
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self.ordered_adjs = {}
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self.ref = defaultdict(lambda: None)
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self.side = defaultdict(lambda: 1)
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# stack of conflict pairs
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self.S = []
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self.stack_bottom = {}
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self.lowpt_edge = {}
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self.left_ref = {}
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self.right_ref = {}
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self.embedding = PlanarEmbedding()
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def lr_planarity(self):
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"""Execute the LR planarity test.
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Returns
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-------
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embedding : dict
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If the graph is planar an embedding is returned. Otherwise None.
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"""
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if self.G.order() > 2 and self.G.size() > 3 * self.G.order() - 6:
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# graph is not planar
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return None
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# make adjacency lists for dfs
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for v in self.G:
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self.adjs[v] = list(self.G[v])
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# orientation of the graph by depth first search traversal
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for v in self.G:
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if self.height[v] is None:
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self.height[v] = 0
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self.roots.append(v)
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self.dfs_orientation(v)
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# Free no longer used variables
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self.G = None
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self.lowpt2 = None
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self.adjs = None
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# testing
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for v in self.DG: # sort the adjacency lists by nesting depth
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# note: this sorting leads to non linear time
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self.ordered_adjs[v] = sorted(
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self.DG[v], key=lambda x: self.nesting_depth[(v, x)]
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)
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for v in self.roots:
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if not self.dfs_testing(v):
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return None
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# Free no longer used variables
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self.height = None
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self.lowpt = None
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self.S = None
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self.stack_bottom = None
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self.lowpt_edge = None
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for e in self.DG.edges:
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self.nesting_depth[e] = self.sign(e) * self.nesting_depth[e]
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self.embedding.add_nodes_from(self.DG.nodes)
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for v in self.DG:
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# sort the adjacency lists again
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self.ordered_adjs[v] = sorted(
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self.DG[v], key=lambda x: self.nesting_depth[(v, x)]
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)
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# initialize the embedding
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previous_node = None
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for w in self.ordered_adjs[v]:
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self.embedding.add_half_edge_cw(v, w, previous_node)
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previous_node = w
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# Free no longer used variables
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self.DG = None
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self.nesting_depth = None
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self.ref = None
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# compute the complete embedding
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for v in self.roots:
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self.dfs_embedding(v)
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# Free no longer used variables
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self.roots = None
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self.parent_edge = None
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self.ordered_adjs = None
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self.left_ref = None
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self.right_ref = None
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self.side = None
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return self.embedding
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def lr_planarity_recursive(self):
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"""Recursive version of :meth:`lr_planarity`."""
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if self.G.order() > 2 and self.G.size() > 3 * self.G.order() - 6:
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# graph is not planar
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return None
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# orientation of the graph by depth first search traversal
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for v in self.G:
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if self.height[v] is None:
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self.height[v] = 0
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self.roots.append(v)
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self.dfs_orientation_recursive(v)
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# Free no longer used variable
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self.G = None
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# testing
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for v in self.DG: # sort the adjacency lists by nesting depth
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# note: this sorting leads to non linear time
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self.ordered_adjs[v] = sorted(
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self.DG[v], key=lambda x: self.nesting_depth[(v, x)]
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)
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for v in self.roots:
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if not self.dfs_testing_recursive(v):
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return None
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for e in self.DG.edges:
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self.nesting_depth[e] = self.sign_recursive(e) * self.nesting_depth[e]
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self.embedding.add_nodes_from(self.DG.nodes)
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for v in self.DG:
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# sort the adjacency lists again
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self.ordered_adjs[v] = sorted(
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self.DG[v], key=lambda x: self.nesting_depth[(v, x)]
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)
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# initialize the embedding
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previous_node = None
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for w in self.ordered_adjs[v]:
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self.embedding.add_half_edge_cw(v, w, previous_node)
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previous_node = w
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# compute the complete embedding
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for v in self.roots:
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self.dfs_embedding_recursive(v)
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return self.embedding
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def dfs_orientation(self, v):
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"""Orient the graph by DFS, compute lowpoints and nesting order.
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"""
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# the recursion stack
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dfs_stack = [v]
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# index of next edge to handle in adjacency list of each node
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ind = defaultdict(lambda: 0)
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# boolean to indicate whether to skip the initial work for an edge
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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]
|
||
|
|
|
||
|
|
|
||
|
|
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()
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
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"]
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
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
|