from collections import defaultdict import networkx as nx __all__ = ["check_planarity", "is_planar", "PlanarEmbedding"] @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] @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] class PlanarEmbedding(nx.DiGraph): """Represents a planar graph with its planar embedding. The planar embedding is given by a `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 `_. 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() """ 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." ) 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 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 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) 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 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"] 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) 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) 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) 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) 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) 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) 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) 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) 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 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