"""
Module for rigidity related graph properties.
"""
from __future__ import annotations
from copy import deepcopy
from itertools import combinations
from typing import List, Union, Iterable
import networkx as nx
import matplotlib.pyplot as plt
from sympy import Matrix
import math
import distinctipy
from pyrigi.data_type import Vertex, Edge, Point
from pyrigi.misc import doc_category, generate_category_tables
from pyrigi.exception import LoopError
import pyrigi._pebble_digraph
[docs]
class Graph(nx.Graph):
"""
Class representing a graph.
One option for *incoming_graph_data* is a list of edges.
See :class:`networkx.Graph` for the other input formats
or use class methods :meth:`~Graph.from_vertices_and_edges`
or :meth:`~Graph.from_vertices` when specifying the vertex set is needed.
Examples
--------
>>> from pyrigi import Graph
>>> G = Graph([(0,1), (1,2), (2,3), (0,3)])
>>> print(G)
Graph with vertices [0, 1, 2, 3] and edges [[0, 1], [0, 3], [1, 2], [2, 3]]
>>> G = Graph()
>>> G.add_vertices([0,2,5,7,'a'])
>>> G.add_edges([(0,7), (2,5)])
>>> print(G)
Graph with vertices [0, 2, 5, 7, 'a'] and edges [[0, 7], [2, 5]]
TODO
----
Implement an alias for plotting.
Graphical output in Jupyter.
Graph names.
Describe in the documentation when an output
of a randomized algorithm is guaranteed to be correct.
Switch from parameter `combinatorial=True/False`
to `algorithm='combinatorial'/'randomized'...`
METHODS
Notes
-----
This class inherits the class :class:`networkx.Graph`.
Some of the inherited methods are for instance:
.. autosummary::
networkx.Graph.add_edge
Many of the :doc:`NetworkX <networkx:index>` algorithms are implemented as functions,
namely, a :class:`Graph` instance has to be passed as the first parameter.
See for instance:
.. autosummary::
~networkx.classes.function.degree
~networkx.classes.function.neighbors
~networkx.classes.function.non_neighbors
~networkx.classes.function.subgraph
~networkx.classes.function.edge_subgraph
~networkx.classes.function.edges
~networkx.algorithms.connectivity.edge_augmentation.is_k_edge_connected
~networkx.algorithms.components.is_connected
~networkx.algorithms.tree.recognition.is_tree
The following links give more information on :class:`networkx.Graph` functionality:
- :doc:`Graph display <networkx:reference/drawing>`
- :doc:`Directed Graphs <networkx:reference/classes/digraph>`
- :doc:`Linear Algebra on Graphs <networkx:reference/linalg>`
- :doc:`A Database of some Graphs <networkx:reference/generators>`
- :doc:`Reading and Writing Graphs <networkx:reference/readwrite/index>`
- :doc:`Converting to and from other Data Formats <networkx:reference/convert>`
"""
[docs]
def __str__(self) -> str:
"""
Return the string representation.
"""
return (
self.__class__.__name__
+ f" with vertices {self.vertex_list()} and edges {self.edge_list()}"
)
[docs]
def __repr__(self) -> str:
"""
Return a representation.
"""
return self.__str__()
[docs]
def __eq__(self, other: Graph):
"""
Return whether the other graph has the same vertices and edges.
Examples
--------
>>> from pyrigi import Graph
>>> G = Graph([[1,2]])
>>> H = Graph([[2,1]])
>>> G == H
True
Note
----
:func:`~networkx.utils.misc.graphs_equal(self, other)`
behaves strangely, hence it is not used.
"""
if (
self.number_of_edges() != other.number_of_edges()
or self.number_of_nodes() != other.number_of_nodes()
):
return False
for v in self.nodes:
if v not in other.nodes:
return False
for e in self.edges:
if not other.has_edge(*e):
return False
return True
[docs]
@classmethod
@doc_category("Class methods")
def from_vertices_and_edges(
cls, vertices: List[Vertex], edges: List[Edge]
) -> Graph:
"""
Create a graph from a list of vertices and edges.
Parameters
----------
vertices
edges
Examples
--------
>>> Graph.from_vertices_and_edges([0, 1, 2, 3], [])
Graph with vertices [0, 1, 2, 3] and edges []
>>> Graph.from_vertices_and_edges([0, 1, 2, 3], [[0, 1], [0, 2], [1, 3]])
Graph with vertices [0, 1, 2, 3] and edges [[0, 1], [0, 2], [1, 3]]
>>> Graph.from_vertices_and_edges(['a', 'b', 'c', 'd'], [['a','c'], ['a', 'd']])
Graph with vertices ['a', 'b', 'c', 'd'] and edges [['a', 'c'], ['a', 'd']]
"""
G = Graph()
G.add_nodes_from(vertices)
G._check_edge_format_list(edges)
G.add_edges(edges)
return G
[docs]
@classmethod
@doc_category("Class methods")
def from_vertices(cls, vertices: List[Vertex]) -> Graph:
"""
Create a graph with no edges from a list of vertices.
Examples
--------
>>> from pyrigi import Graph
>>> G = Graph.from_vertices([3, 1, 7, 2, 12, 3, 0])
>>> G
Graph with vertices [0, 1, 2, 3, 7, 12] and edges []
"""
return Graph.from_vertices_and_edges(vertices, [])
[docs]
@classmethod
@doc_category("Class methods")
def CompleteOnVertices(cls, vertices: List[Vertex]) -> Graph:
"""
Generate a complete graph on ``vertices``.
Examples
--------
>>> Graph.CompleteOnVertices([0, 1, 2, 3, 4])
Graph with vertices [0, 1, 2, 3, 4] and edges [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]
>>> Graph.CompleteOnVertices(['a', 'b', 'c', 'd'])
Graph with vertices ['a', 'b', 'c', 'd'] and edges [['a', 'b'], ['a', 'c'], ['a', 'd'], ['b', 'c'], ['b', 'd'], ['c', 'd']]
""" # noqa: E501
edges = list(combinations(vertices, 2))
return Graph.from_vertices_and_edges(vertices, edges)
def _check_edge_format(self, input_pair: Edge) -> None:
"""
Check if an input_pair is a pair of distinct vertices of the graph.
"""
if (
not (isinstance(input_pair, tuple) or isinstance(input_pair, list))
or not len(input_pair) == 2
):
raise TypeError(
f"The input {input_pair} must be a tuple or list of length 2."
)
if not input_pair[0] in self.nodes or not input_pair[1] in self.nodes:
raise ValueError(
f"The elements of the pair {input_pair} are not vertices of the graph."
)
if input_pair[0] == input_pair[1]:
raise LoopError("The input {input_pair} must be two distinct vertices.")
def _check_edge(self, edge: Edge, vertices: List[Vertex] = None) -> None:
"""
Check if the given input is an edge of the graph with endvertices in vertices.
Parameters
----------
edge:
an edge to be checked
vertices:
Check if the endvertices of the edge are contained in the list ``vertices``.
If None, the function considers all vertices of the graph.
"""
self._check_edge_format(edge)
if vertices and (not edge[0] in vertices or not edge[1] in vertices):
raise ValueError(
f"The elements of the edge {edge} are not among vertices {vertices}."
)
if not self.has_edge(edge[0], edge[1]):
raise ValueError(f"Edge {edge} is not contained in the graph.")
def _check_edge_list(
self, edges: List[Edge], vertices: List[Vertex] = None
) -> None:
"""
Apply _check_edge to all edges in a list.
Parameters
----------
edges:
a list of edges to be checked
vertices:
Check if the endvertices of the edges are contained in the list ``vertices``.
If None (default), the function considers all vertices of the graph.
"""
for edge in edges:
self._check_edge(edge, vertices)
def _check_edge_format_list(self, pairs: List[Edge]) -> None:
"""
Apply _check_edge_format to all pairs in a list.
Parameters
----------
pairs:
a list of pairs to be checked
"""
for pair in pairs:
self._check_edge_format(pair)
[docs]
@doc_category("Attribute getters")
def vertex_list(self) -> List[Vertex]:
"""
Return the list of vertices.
The output is sorted if possible,
otherwise, the internal order is used instead.
Examples
--------
>>> G = Graph.from_vertices_and_edges([2, 0, 3, 1], [[0, 1], [0, 2], [0, 3]])
>>> G.vertex_list()
[0, 1, 2, 3]
>>> G = Graph.from_vertices(['c', 'a', 'b'])
>>> G.vertex_list()
['a', 'b', 'c']
>>> G = Graph.from_vertices(['b', 1, 'a']) # incomparable vertices
>>> G.vertex_list()
['b', 1, 'a']
"""
try:
return sorted(self.nodes)
except BaseException:
return list(self.nodes)
[docs]
@doc_category("Attribute getters")
def edge_list(self) -> List[Edge]:
"""
Return the list of edges.
The output is sorted if possible,
otherwise, the internal order is used instead.
Examples
--------
>>> G = Graph([[0, 3], [3, 1], [0, 1], [2, 0]])
>>> G.edge_list()
[[0, 1], [0, 2], [0, 3], [1, 3]]
>>> G = Graph.from_vertices(['a', 'c', 'b'])
>>> G.edge_list()
[]
>>> G = Graph([['c', 'b'], ['b', 'a']])
>>> G.edge_list()
[['a', 'b'], ['b', 'c']]
>>> G = Graph([['c', 1], [2, 'a']]) # incomparable vertices
>>> G.edge_list()
[('c', 1), (2, 'a')]
"""
try:
return sorted([sorted(e) for e in self.edges])
except BaseException:
return list(self.edges)
[docs]
@doc_category("Graph manipulation")
def delete_vertex(self, vertex: Vertex) -> None:
"""Alias for :meth:`networkx.Graph.remove_node`."""
self.remove_node(vertex)
[docs]
@doc_category("Graph manipulation")
def delete_vertices(self, vertices: List[Vertex]) -> None:
"""Alias for :meth:`networkx.Graph.remove_nodes_from`."""
self.remove_nodes_from(vertices)
[docs]
@doc_category("Graph manipulation")
def delete_edge(self, edge: Edge) -> None:
"""Alias for :meth:`networkx.Graph.remove_edge`"""
self.remove_edge(*edge)
[docs]
@doc_category("Graph manipulation")
def delete_edges(self, edges: List[Edge]) -> None:
"""Alias for :meth:`networkx.Graph.remove_edges_from`."""
self.remove_edges_from(edges)
[docs]
@doc_category("Graph manipulation")
def add_vertex(self, vertex: Vertex) -> None:
"""Alias for :meth:`networkx.Graph.add_node`."""
self.add_node(vertex)
[docs]
@doc_category("Graph manipulation")
def add_vertices(self, vertices: List[Vertex]) -> None:
"""Alias for :meth:`networkx.Graph.add_nodes_from`."""
self.add_nodes_from(vertices)
[docs]
@doc_category("Graph manipulation")
def add_edges(self, edges: List[Edge]) -> None:
"""Alias for :meth:`networkx.Graph.add_edges_from`."""
self.add_edges_from(edges)
[docs]
@doc_category("Graph manipulation")
def delete_loops(self) -> None:
"""Removes all the loops from the edges to get a loop free graph."""
self.delete_edges(nx.selfloop_edges(self))
[docs]
@doc_category("General graph theoretical properties")
def vertex_connectivity(self) -> int:
"""Alias for :func:`networkx.algorithms.connectivity.connectivity.node_connectivity`.""" # noqa: E501
return nx.node_connectivity(self)
[docs]
@doc_category("General graph theoretical properties")
def degree_sequence(self, vertex_order: List[Vertex] = None) -> list[int]:
"""
Return a list of degrees of the vertices of the graph.
Parameters
----------
vertex_order:
By listing vertices in the preferred order, the degree_sequence
can be computed in a way the user expects. If no vertex order is
provided, :meth:`~.Graph.vertex_list()` is used.
Examples
--------
>>> G = Graph([(0,1), (1,2)])
>>> G.degree_sequence()
[1, 2, 1]
"""
if vertex_order is None:
vertex_order = self.vertex_list()
else:
if not set(self.nodes) == set(
vertex_order
) or not self.number_of_nodes() == len(vertex_order):
raise IndexError(
"The vertex_order must contain the same vertices as the graph!"
)
return [self.degree(v) for v in vertex_order]
[docs]
@doc_category("General graph theoretical properties")
def min_degree(self) -> int:
"""
Return the minimum of the vertex degrees.
Examples
--------
>>> G = Graph([(0,1), (1,2)])
>>> G.min_degree()
1
"""
return min([self.degree(v) for v in self.nodes])
[docs]
@doc_category("General graph theoretical properties")
def max_degree(self) -> int:
"""
Return the maximum of the vertex degrees.
Examples
--------
>>> G = Graph([(0,1), (1,2)])
>>> G.max_degree()
2
"""
return max([self.degree(v) for v in self.nodes])
@staticmethod
@doc_category("Sparseness")
def _pebble_values_are_correct(K: int, L: int) -> bool:
r"""
Check if K and L satisfy pebble game conditions.
K and L need to be integers that satisfy the conditions
K > 0, L >= 0 and L < 2K
"""
if not (isinstance(K, int) and isinstance(L, int)):
return False
if K <= 0 or L < 0 or L >= 2 * K:
return False
return True
@doc_category("Sparseness")
def _build_pebble_digraph(self, K: int, L: int) -> None:
r"""
Build and save the pebble digraph from scratch.
Adds edges one-by-one, as long as it can.
Discard edges that are not :prf:ref:`(K, L)-independent <def-kl-sparse-tight>`
from the rest of the graph.
"""
if not self._pebble_values_are_correct(K, L):
raise TypeError(
"K and L need to be integers that satisfy the conditions of\
K > 0, L >= 0 and L < 2K."
)
dir_graph = pyrigi._pebble_digraph.PebbleDiGraph(K, L)
dir_graph.add_nodes_from(self.nodes)
for edge in self.edges:
u, v = edge[0], edge[1]
dir_graph.add_edge_maintaining_digraph(u, v)
self._pebble_digraph = dir_graph
[docs]
@doc_category("Sparseness")
def spanning_sparse_subgraph(
self, K: int, L: int, use_precomputed_pebble_digraph: bool = False
) -> Graph:
r"""
Return a maximal :prf:ref:`(K, L)-sparse <def-kl-sparse-tight>` subgraph.
Based on the directed graph calculated by the pebble game algorithm, return
a maximal :prf:ref:`(K, L)-sparse <def-kl-sparse-tight>` of the graph.
There are multiple possible maximal (K, L)-sparse subgraphs, all of which have
the same number of edges.
Parameters
----------
K:
L:
use_precomputed_pebble_digraph:
If ``True``, the pebble digraph present in the cache is used.
If ``False``, recompute the pebble digraph.
Use ``True`` only if you are certain that the pebble game digraph
is consistent with the graph.
"""
if (
not use_precomputed_pebble_digraph
or K != self._pebble_digraph.K
or L != self._pebble_digraph.L
):
self._build_pebble_digraph(K, L)
return self._pebble_digraph.to_undirected()
@doc_category("Sparseness")
def _is_pebble_digraph_sparse(
self, K: int, L: int, use_precomputed_pebble_digraph: bool = False
) -> bool:
"""
Check whether the pebble digraph has the same number of edges as the graph.
Parameters
----------
K:
L:
use_precomputed_pebble_digraph:
If ``True``, the pebble digraph present in the cache is used.
If ``False``, recompute the pebble digraph.
Use ``True`` only if you are certain that the pebble game digraph
is consistent with the graph.
"""
if (
not use_precomputed_pebble_digraph
or K != self._pebble_digraph.K
or L != self._pebble_digraph.L
):
self._build_pebble_digraph(K, L)
# all edges are in fact inside the pebble digraph
return self.number_of_edges() == self._pebble_digraph.number_of_edges()
[docs]
@doc_category("Sparseness")
def is_sparse(
self,
K: int,
L: int,
algorithm: str = "default",
use_precomputed_pebble_digraph: bool = False,
) -> bool:
r"""
Check whether the graph is :prf:ref:`(K, L)-sparse <def-kl-sparse-tight>`.
Parameters
----------
K:
L:
algorithm:
"pebble" or "subgraph".
If "pebble", the function uses the pebble game algorithm to check
for sparseness. If "subgraph", it uses the subgraph method.
If not specified, it defaults to "pebble" whenever possible,
otherwise "subgraph".
use_precomputed_pebble_digraph:
If ``True``, the pebble digraph present in the cache is used.
If ``False``, recompute the pebble digraph.
Use ``True`` only if you are certain that the pebble game digraph
is consistent with the graph.
TODO
----
examples, tests for other cases than (2,3)
"""
if not (isinstance(K, int) and isinstance(L, int)):
raise TypeError("K and L need to be integers!")
if algorithm == "pebble":
if self._pebble_values_are_correct(K, L):
return self._is_pebble_digraph_sparse(
K, L, use_precomputed_pebble_digraph=use_precomputed_pebble_digraph
)
else:
raise ValueError(
"K and L with pebble algorithm need to satisfy the\
conditions of K > 0, 0 <= L < 2K."
)
if algorithm == "subgraph":
for j in range(K, self.number_of_nodes() + 1):
for vertex_set in combinations(self.nodes, j):
G = self.subgraph(vertex_set)
if G.number_of_edges() > K * G.number_of_nodes() - L:
return False
return True
if algorithm == "default":
if self._pebble_values_are_correct(K, L):
# use "pebble" if possible
algorithm = "pebble"
else:
# otherwise use "subgraph"
algorithm = "subgraph"
return self.is_sparse(
K,
L,
algorithm,
use_precomputed_pebble_digraph=use_precomputed_pebble_digraph,
)
# reaching this position means that the algorithm is unknown
raise ValueError(
f"If specified, the value of the algorithm parameter must be one of "
f'"pebble", "subgraph", or "default". Instead, it is {algorithm}.'
)
[docs]
@doc_category("Sparseness")
def is_tight(
self,
K: int,
L: int,
algorithm: str = "default",
use_precomputed_pebble_digraph: bool = False,
) -> bool:
r"""
Check whether the graph is :prf:ref:`(K, L)-tight <def-kl-sparse-tight>`.
Parameters
----------
K:
L:
algorithm:
"pebble" or "subgraph".
If "pebble", the function uses the pebble game algorithm to check
for sparseness. If "subgraph", it uses the subgraph method.
If not specified, it defaults to "pebble".
use_precomputed_pebble_digraph:
If ``True``, the pebble digraph present in the cache is used.
If ``False``, recompute the pebble digraph.
Use ``True`` only if you are certain that the pebble game digraph
is consistent with the graph.
TODO
----
examples, tests for other cases than (2,3)
"""
return (
self.is_sparse(
K,
L,
algorithm,
use_precomputed_pebble_digraph=use_precomputed_pebble_digraph,
)
and self.number_of_edges() == K * self.number_of_nodes() - L
)
[docs]
@doc_category("Graph manipulation")
def zero_extension(
self,
vertices: List[Vertex],
new_vertex: Vertex = None,
dim: int = 2,
inplace: bool = False,
) -> Graph:
"""
Return a :prf:ref:`dim-dimensional 0-extension <def-k-extension>`.
Parameters
----------
vertices:
A new vertex will be connected to these vertices.
All the vertices must be contained in the graph
and there must be ``dim`` of them.
new_vertex:
Newly added vertex will be named according to this parameter.
If None, the name will be set as the lowest possible integer value
greater or equal than the number of nodes.
dim:
The dimension in which the k-extension is created.
inplace:
If True, the graph will be modified,
otherwise a new modified graph will be created,
while the original graph remains unchanged (default).
Examples
--------
>>> import pyrigi.graphDB as graphs
>>> G = graphs.Complete(3)
>>> G
Graph with vertices [0, 1, 2] and edges [[0, 1], [0, 2], [1, 2]]
>>> G.zero_extension([0, 2])
Graph with vertices [0, 1, 2, 3] and edges [[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]
>>> G.zero_extension([0, 2], 5)
Graph with vertices [0, 1, 2, 5] and edges [[0, 1], [0, 2], [0, 5], [1, 2], [2, 5]]
>>> G
Graph with vertices [0, 1, 2] and edges [[0, 1], [0, 2], [1, 2]]
>>> G.zero_extension([0, 1, 2], 5, dim=3, inplace=True);
Graph with vertices [0, 1, 2, 5] and edges [[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [2, 5]]
>>> G
Graph with vertices [0, 1, 2, 5] and edges [[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [2, 5]]
""" # noqa: E501
return self.k_extension(0, vertices, [], new_vertex, dim, inplace)
[docs]
@doc_category("Graph manipulation")
def one_extension(
self,
vertices: List[Vertex],
edge: Edge,
new_vertex: Vertex = None,
dim: int = 2,
inplace: bool = False,
) -> Graph:
"""
Return a :prf:ref:`dim-dimensional 1-extension <def-k-extension>`.
Parameters
----------
vertices:
A new vertex will be connected to these vertices.
All the vertices must be contained in the graph
and there must be ``dim + 1`` of them.
edge:
An edge with endvertices from the list ``vertices`` that will be deleted.
The edge must be contained in the graph.
new_vertex:
Newly added vertex will be named according to this parameter.
If None, the name will be set as the lowest possible integer value
greater or equal than the number of nodes.
dim:
The dimension in which the k-extension is created.
inplace:
If True, the graph will be modified,
otherwise a new modified graph will be created,
while the original graph remains unchanged (default).
Examples
--------
>>> import pyrigi.graphDB as graphs
>>> G = graphs.Complete(3)
>>> G
Graph with vertices [0, 1, 2] and edges [[0, 1], [0, 2], [1, 2]]
>>> G.one_extension([0, 1, 2], [0, 1])
Graph with vertices [0, 1, 2, 3] and edges [[0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]
>>> G
Graph with vertices [0, 1, 2] and edges [[0, 1], [0, 2], [1, 2]]
>>> G = graphs.ThreePrism()
>>> G
Graph with vertices [0, 1, 2, 3, 4, 5] and edges [[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 5], [3, 4], [3, 5], [4, 5]]
>>> G.one_extension([0, 1], [0, 1], dim=1)
Graph with vertices [0, 1, 2, 3, 4, 5, 6] and edges [[0, 2], [0, 3], [0, 6], [1, 2], [1, 4], [1, 6], [2, 5], [3, 4], [3, 5], [4, 5]]
>>> G = graphs.CompleteBipartite(3, 2)
>>> G
Graph with vertices [0, 1, 2, 3, 4] and edges [[0, 3], [0, 4], [1, 3], [1, 4], [2, 3], [2, 4]]
>>> G.one_extension([0, 1, 2, 3, 4], [0, 3], dim=4, inplace = True)
Graph with vertices [0, 1, 2, 3, 4, 5] and edges [[0, 4], [0, 5], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 5], [4, 5]]
>>> G
Graph with vertices [0, 1, 2, 3, 4, 5] and edges [[0, 4], [0, 5], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 5], [4, 5]]
""" # noqa: E501
return self.k_extension(1, vertices, [edge], new_vertex, dim, inplace)
[docs]
@doc_category("Graph manipulation")
def k_extension(
self,
k: int,
vertices: List[Vertex],
edges: List[Edge],
new_vertex: Vertex = None,
dim: int = 2,
inplace: bool = False,
) -> Graph:
"""
Return a :prf:ref:`dim-dimensional k-extension <def-k-extension>`.
Parameters
----------
k
vertices:
A new vertex will be connected to these vertices.
All the vertices must be contained in the graph
and there must be ``dim + k`` of them.
edges:
A list of edges that will be deleted.
The endvertices of all the edges must be contained
in the list ``vertices``.
The edges must be contained in the graph and there must be k of them.
new_vertex:
Newly added vertex will be named according to this parameter.
If None, the name will be set as the lowest possible integer value
greater or equal than the number of nodes.
dim:
The dimension in which the k-extension is created.
inplace:
If True, the graph will be modified,
otherwise a new modified graph will be created,
while the original graph remains unchanged (default).
Notes
-----
See also :meth:`~Graph.zero_extension` and :meth:`~Graph.one_extension`.
Examples
--------
>>> import pyrigi.graphDB as graphs
>>> G = graphs.Complete(5)
>>> G
Graph with vertices [0, 1, 2, 3, 4] and edges [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]
>>> G.k_extension(2, [0, 1, 2, 3], [[0, 1], [0,2]])
Graph with vertices [0, 1, 2, 3, 4, 5] and edges [[0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5]]
>>> G
Graph with vertices [0, 1, 2, 3, 4] and edges [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]
>>> G = graphs.Complete(5)
>>> G
Graph with vertices [0, 1, 2, 3, 4] and edges [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]
>>> G.k_extension(2, [0, 1, 2, 3, 4], [[0, 1], [0,2]], dim = 3)
Graph with vertices [0, 1, 2, 3, 4, 5] and edges [[0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]]
>>> G = graphs.Path(6)
>>> G
Graph with vertices [0, 1, 2, 3, 4, 5] and edges [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]
>>> G.k_extension(2, [0, 1, 2], [[0, 1], [1,2]], dim = 1, inplace = True)
Graph with vertices [0, 1, 2, 3, 4, 5, 6] and edges [[0, 6], [1, 6], [2, 3], [2, 6], [3, 4], [4, 5]]
>>> G
Graph with vertices [0, 1, 2, 3, 4, 5, 6] and edges [[0, 6], [1, 6], [2, 3], [2, 6], [3, 4], [4, 5]]
""" # noqa: E501
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
for vertex in vertices:
if vertex not in self.nodes:
raise ValueError(f"Vertex {vertex} is not contained in the graph")
if len(set(vertices)) != dim + k:
raise ValueError(
f"List of vertices must contain {dim + k} distinct vertices"
)
self._check_edge_list(edges, vertices)
if len(edges) != k:
raise ValueError(f"List of edges must contain {k} distinct edges")
if new_vertex is None:
candidate = self.number_of_nodes()
while candidate in self.nodes:
candidate += 1
new_vertex = candidate
if new_vertex in self.nodes:
raise ValueError(f"Vertex {new_vertex} is already a vertex of the graph!")
G = self
if not inplace:
G = deepcopy(self)
G.remove_edges_from(edges)
for vertex in vertices:
G.add_edge(vertex, new_vertex)
return G
[docs]
@doc_category("Graph manipulation")
def all_k_extensions(
self,
k: int,
dim: int = 2,
only_non_isomorphic: bool = False,
) -> Iterable[Graph]:
"""
Return an iterator over all possible
:prf:ref:`dim-dimensional k-extensions <def-k-extension>`.
Parameters
----------
k
dim
only_non_isomorphic:
If True, only one graph per isomorphism class is included.
Examples
--------
>>> import pyrigi.graphDB as graphs
>>> G = graphs.Complete(3)
>>> type(G.all_k_extensions(0))
<class 'generator'>
>>> len(list(G.all_k_extensions(0)))
3
>>> len(list(G.all_k_extensions(0, only_non_isomorphic=True)))
1
>>> len(list(graphs.Diamond().all_k_extensions(1, 2, only_non_isomorphic=True)))
2
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if self.number_of_nodes() < (dim + k):
raise ValueError(
f"The number of nodes in the graph needs to be "
f"greater or equal than {dim + k}!"
)
if self.number_of_edges() < k:
raise ValueError(
f"The number of edges in the graph needs to be greater or equal than {k}!"
)
solutions = []
for edges in combinations(self.edges, k):
s = set(self.nodes)
w = set()
for edge in edges:
s.discard(edge[0])
s.discard(edge[1])
w.add(edge[0])
w.add(edge[1])
if len(w) > (dim + k):
break
w = list(w)
for vertices in combinations(s, dim + k - len(w)):
current = self.k_extension(k, list(vertices) + w, edges, dim=dim)
if only_non_isomorphic:
for other in solutions:
if current.is_isomorphic(other):
break
else:
solutions.append(current)
yield current
else:
yield current
[docs]
@doc_category("Generic rigidity")
def extension_sequence(
self, dim: int = 2, return_solution: bool = False
) -> Union[List[Graph], bool]:
"""
Check the existence of a sequence of
:prf:ref:`0 and 1-extensions <def-k-extension>`.
The method returns whether the graph can be constructed
by a sequence of 0 and 1-extensions starting from an edge.
Parameters
----------
dim:
The dimension in which the extensions are created.
Currently implemented only for ``dim==2``.
return_solution:
If False, a boolean value indicating if the graph can be
created by a sequence of extensions is returned.
If True, an extension sequence of graphs that creates the graph
is returned, or None if no such extension sequence exists.
Examples
--------
>>> import pyrigi.graphDB as graphs
>>> G = graphs.ThreePrism()
>>> G
Graph with vertices [0, 1, 2, 3, 4, 5] and edges [[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 5], [3, 4], [3, 5], [4, 5]]
>>> G.extension_sequence()
True
>>> G = graphs.CompleteBipartite(1, 2)
>>> G
Graph with vertices [0, 1, 2] and edges [[0, 1], [0, 2]]
>>> G.extension_sequence()
False
>>> G = graphs.Complete(3)
>>> G
Graph with vertices [0, 1, 2] and edges [[0, 1], [0, 2], [1, 2]]
>>> G.extension_sequence(return_solution=True)
[Graph with vertices [1, 2] and edges [[1, 2]], Graph with vertices [0, 1, 2] and edges [[0, 1], [0, 2], [1, 2]]]
>>> G = graphs.Diamond()
>>> G
Graph with vertices [0, 1, 2, 3] and edges [[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]
>>> G.extension_sequence(return_solution=True)
[Graph with vertices [2, 3] and edges [[2, 3]], Graph with vertices [0, 2, 3] and edges [[0, 2], [0, 3], [2, 3]], Graph with vertices [0, 1, 2, 3] and edges [[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]]
""" # noqa: E501
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if not dim == 2:
raise NotImplementedError()
if not self.number_of_edges() == 2 * self.number_of_nodes() - 3:
return None if return_solution else False
if self.number_of_nodes() == 2:
return [self] if return_solution else True
degrees = sorted(self.degree, key=lambda node: node[1])
if degrees[0][1] < 2 or degrees[0][1] > 3:
return None if return_solution else False
if degrees[0][1] == 2:
G = deepcopy(self)
G.remove_node(degrees[0][0])
branch = G.extension_sequence(dim, return_solution)
if return_solution:
if branch is not None:
return branch + [self]
return None
return branch
if degrees[0][1] == 3:
neighbors = list(self.neighbors(degrees[0][0]))
G = deepcopy(self)
G.remove_node(degrees[0][0])
for i, j in [[0, 1], [0, 2], [1, 2]]:
if not G.has_edge(neighbors[i], neighbors[j]):
G.add_edge(neighbors[i], neighbors[j])
branch = G.extension_sequence(dim, return_solution)
if return_solution and branch is not None:
return branch + [self]
elif branch:
return True
G.remove_edge(neighbors[i], neighbors[j])
return None if return_solution else False
[docs]
@doc_category("Generic rigidity")
def is_vertex_redundantly_rigid(
self, dim: int = 2, combinatorial: bool = True
) -> bool:
"""
Check whether the graph is :prf:ref:`vertex redundantly (generically) dim-rigid
<def-redundantly-rigid-graph>`.
See :meth:`.is_k_vertex_redundantly_rigid` (using k = 1) for details.
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
return self.is_k_vertex_redundantly_rigid(1, dim, combinatorial)
[docs]
@doc_category("Generic rigidity")
def is_k_vertex_redundantly_rigid(
self, k: int, dim: int = 2, combinatorial: bool = True
) -> bool:
"""
Check whether the graph is :prf:ref:`k-vertex redundantly (generically) dim-rigid
<def-redundantly-rigid-graph>`.
Preliminary checks from
:prf:ref:`thm-k-vertex-redundant-edge-bound-general`,
:prf:ref:`thm-k-vertex-redundant-edge-bound-general2`,
:prf:ref:`thm-1-vertex-redundant-edge-bound-dim2`,
:prf:ref:`thm-2-vertex-redundant-edge-bound-dim2`
:prf:ref:`thm-k-vertex-redundant-edge-bound-dim2`,
:prf:ref:`thm-3-vertex-redundant-edge-bound-dim3`,
:prf:ref:`thm-k-vertex-redundant-edge-bound-dim3`
... are used
Examples
--------
>>> G = Graph([[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]])
>>> G.is_k_vertex_redundantly_rigid(1, 2)
True
>>> G.is_k_vertex_redundantly_rigid(2, 2)
False
>>> G = Graph([[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 4], [3, 4]])
>>> G.is_k_vertex_redundantly_rigid(1, 2)
False
""" # noqa: E501
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if not isinstance(k, int):
raise TypeError(f"k needs to be a nonnegative integer, but is {k}!")
if nx.number_of_selfloops(self) > 0:
raise LoopError()
n = self.number_of_nodes()
m = self.number_of_edges()
if n >= dim + k + 1 and self.min_degree() < dim + k:
return False
if dim == 1:
return self.vertex_connectivity() >= k + 1
if (
dim == 2
and (
# edge bound from :prf:ref:`thm-1-vertex-redundant-edge-bound-dim2`
(k == 1 and n >= 5 and m < 2 * n - 1)
or
# edge bound from :prf:ref:`thm-2-vertex-redundant-edge-bound-dim2`
(k == 2 and n >= 6 and m < 2 * n + 2)
or
# edge bound from :prf:ref:`thm-k-vertex-redundant-edge-bound-dim2`
(k >= 3 and n >= 6 * (k + 1) + 23 and m < ((k + 2) * n + 1) // 2)
)
) or (
dim == 3
and (
# edge bound from :prf:ref:`thm-3-vertex-redundant-edge-bound-dim3`
(k == 3 and n >= 15 and m < 3 * n + 5)
or
# edge bound from :prf:ref:`thm-k-vertex-redundant-edge-bound-dim3`
(
k >= 4
and n >= 12 * (k + 1) + 10
and n % 2 == 0
and m < ((k + 3) * n + 1) // 2
)
)
):
return False
# edge bound from :prf:ref:`thm-k-vertex-redundant-edge-bound-general`
if (
#
n >= dim * dim + dim + k + 1
and m
< dim * n - math.comb(dim + 1, 2) + k * dim + max(0, k - (dim + 1) // 2)
):
return False
# edge bound from :prf:ref:`thm-vertex-redundant-edge-bound-general2`
if k >= dim + 1 and n >= dim + k + 1 and m < ((dim + k) * n + 1) // 2:
return False
# in all other cases check by definition
G = deepcopy(self)
for vertex_set in combinations(self.nodes, k):
adj = [[v, list(G.neighbors(v))] for v in vertex_set]
G.delete_vertices(vertex_set)
if not G.is_rigid(dim, combinatorial):
return False
# add vertices and edges back
G.add_vertices(vertex_set)
for v, neighbors in adj:
for neighbor in neighbors:
G.add_edge(v, neighbor)
return True
[docs]
@doc_category("Generic rigidity")
def is_min_vertex_redundantly_rigid(
self, dim: int = 2, combinatorial: bool = True
) -> bool:
"""
Check whether the graph is
:prf:ref:`minimally vertex redundantly (generically) dim-rigid
<def-min-redundantly-rigid-graph>`.
See :meth:`.is_min_k_vertex_redundantly_rigid` (using k = 1) for details.
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
return self.is_min_k_vertex_redundantly_rigid(1, dim, combinatorial)
[docs]
@doc_category("Generic rigidity")
def is_min_k_vertex_redundantly_rigid(
self, k: int, dim: int = 2, combinatorial: bool = True
) -> bool:
"""
Check whether the graph is :prf:ref:`minimally k-vertex redundantly (generically) dim-rigid
<def-redundantly-rigid-graph>`.
Preliminary checks from
:prf:ref:`thm-minimal-k-vertex-redundant-upper-edge-bound`,
:prf:ref:`thm-minimal-k-vertex-redundant-upper-edge-bound-dim1`
are used.
Examples
--------
>>> G = Graph([[0, 3], [0, 4], [0, 5], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]])
>>> G.is_min_k_vertex_redundantly_rigid(1, 2)
True
>>> G.is_min_k_vertex_redundantly_rigid(2, 2)
False
>>> G = Graph([[0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 4], [2, 5], [3, 4], [3, 5]])
>>> G.is_k_vertex_redundantly_rigid(1, 2)
True
>>> G.is_min_k_vertex_redundantly_rigid(1, 2)
False
""" # noqa: E501
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if not isinstance(k, int):
raise TypeError(f"k needs to be a nonnegative integer, but is {k}!")
if nx.number_of_selfloops(self) > 0:
raise LoopError()
n = self.number_of_nodes()
m = self.number_of_edges()
# edge bound from :prf:ref:`thm-minimal-k-vertex-redundant-upper-edge-bound`
if m > (dim + k) * n - math.comb(dim + k + 1, 2):
return False
# edge bound from :prf:ref:`thm-minimal-k-vertex-redundant-upper-edge-bound-dim1`
if dim == 1:
if n >= 3 * (k + 1) - 1 and m > (k + 1) * n - (k + 1) * (k + 1):
return False
if not self.is_k_vertex_redundantly_rigid(k, dim, combinatorial):
return False
# for the following we need to know that the graph is k-vertex-redundantly rigid
if (
dim == 2
and (
# edge bound from :prf:ref:`thm-1-vertex-redundant-edge-bound-dim2`
(k == 1 and n >= 5 and m == 2 * n - 1)
or
# edge bound from :prf:ref:`thm-2-vertex-redundant-edge-bound-dim2`
(k == 2 and n >= 6 and m == 2 * n + 2)
or
# edge bound from :prf:ref:`thm-k-vertex-redundant-edge-bound-dim2`
(k >= 3 and n >= 6 * (k + 1) + 23 and m == ((k + 2) * n + 1) // 2)
)
) or (
dim == 3
and (
# edge bound from :prf:ref:`thm-3-vertex-redundant-edge-bound-dim3`
(k == 3 and n >= 15 and m == 3 * n + 5)
or
# edge bound from :prf:ref:`thm-k-vertex-redundant-edge-bound-dim3`
(
k >= 4
and n >= 12 * (k + 1) + 10
and n % 2 == 0
and m == ((k + 3) * n + 1) // 2
)
)
):
return True
# edge bound from :prf:ref:`thm-k-vertex-redundant-edge-bound-general`
if (
#
n >= dim * dim + dim + k + 1
and m
== dim * n - math.comb(dim + 1, 2) + k * dim + max(0, k - (dim + 1) // 2)
):
return True
# edge bound from :prf:ref:`thm-vertex-redundant-edge-bound-general2`
if k >= dim + 1 and n >= dim + k + 1 and m == ((dim + k) * n + 1) // 2:
return True
# in all other cases check by definition
G = deepcopy(self)
for edge in self.edge_list():
G.delete_edges([edge])
if G.is_k_vertex_redundantly_rigid(k, dim, combinatorial):
return False
G.add_edges([edge])
return True
[docs]
@doc_category("Generic rigidity")
def is_redundantly_rigid(self, dim: int = 2, combinatorial: bool = True) -> bool:
"""
Check whether the graph is :prf:ref:`redundantly (generically) dim-rigid
<def-redundantly-rigid-graph>`.
See :meth:`.is_k_redundantly_rigid` (using k = 1) for details.
"""
return self.is_k_redundantly_rigid(1, dim, combinatorial)
[docs]
@doc_category("Generic rigidity")
def is_k_redundantly_rigid(
self, k: int, dim: int = 2, combinatorial: bool = True
) -> bool:
"""
Check whether the graph is :prf:ref:`k-redundantly (generically) dim-rigid
<def-redundantly-rigid-graph>`.
Preliminary checks from
:prf:ref:`thm-k-edge-redundant-edge-bound-dim2`,
:prf:ref:`thm-1-edge-redundant-edge-bound-dim2`,
:prf:ref:`thm-2-edge-redundant-edge-bound-dim3`,
:prf:ref:`thm-k-edge-redundant-edge-bound-dim3`,
:prf:ref:`thm-globally-redundant-3connected` and
:prf:ref:`thm-globally-mindeg6-dim2`.
are used
Examples
--------
>>> G = Graph([[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 5], [3, 4], [3, 5], [4, 5]])
>>> G.is_k_redundantly_rigid(1, 2)
True
>>> G = Graph([[0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]])
>>> G.is_k_redundantly_rigid(1, 2)
False
>>> G = Graph([[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]])
>>> G.is_k_redundantly_rigid(2, 2)
True
TODO
----
Improve with pebble games.
""" # noqa: E501
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if not isinstance(k, int):
raise TypeError(f"k needs to be a nonnegative integer, but is {k}!")
if nx.number_of_selfloops(self) > 0:
raise LoopError()
n = self.number_of_nodes()
m = self.number_of_edges()
if m < dim * n - math.comb(dim + 1, 2) + k:
return False
if self.min_degree() < dim + k:
return False
if dim == 1:
return nx.edge_connectivity(self) >= k + 1
# edge bounds
if (
dim == 2
and (
# basic edge bound
(k == 1 and m < 2 * n - 2)
or
# edge bound from :prf:ref:`thm-1-edge-redundant-edge-bound-dim2`
(k == 2 and n >= 5 and m < 2 * n)
or
# edge bound from :prf:ref:`thm-k-edge-redundant-edge-bound-dim2`
(k >= 3 and n >= 6 * (k + 1) + 23 and m < ((k + 2) * n + 1) // 2)
)
) or (
dim == 3
and (
# edge bound from :prf:ref:`thm-2-edge-redundant-edge-bound-dim3`
(k == 2 and n >= 14 and m < 3 * n - 4)
or
# edge bound from :prf:ref:`thm-k-edge-redundant-edge-bound-dim3`
(
k >= 4
and n >= 12 * (k + 1) + 10
and n % 2 == 0
and m < ((k + 3) * n + 1) // 2
)
)
):
return False
# use global rigidity property of :prf:ref:`thm-globally-redundant-3connected`
# and :prf:ref:`thm-globally-mindeg6-dim2`
if dim == 2 and k == 1 and self.vertex_connectivity() >= 6:
return True
# in all other cases check by definition
G = deepcopy(self)
for edge_set in combinations(self.edge_list(), k):
G.delete_edges(edge_set)
if not G.is_rigid(dim, combinatorial):
return False
G.add_edges(edge_set)
return True
[docs]
@doc_category("Generic rigidity")
def is_min_redundantly_rigid(
self, dim: int = 2, combinatorial: bool = True
) -> bool:
"""
Check whether the graph is :prf:ref:`minimally redundantly (generically) dim-rigid
<def-min-redundantly-rigid-graph>`.
See :meth:`.is_min_k_redundantly_rigid` (using k = 1) for details.
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
return self.is_min_k_redundantly_rigid(1, dim, combinatorial)
[docs]
@doc_category("Generic rigidity")
def is_min_k_redundantly_rigid(
self, k: int, dim: int = 2, combinatorial: bool = True
) -> bool:
"""
Check whether the graph is :prf:ref:`minimally k-redundantly (generically) dim-rigid
<def-redundantly-rigid-graph>`.
Preliminary checks from
:prf:ref:`thm-minimal-1-edge-redundant-upper-edge-bound-dim2`
are used.
Examples
--------
>>> G = Graph([[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 4], [3, 4]])
>>> G.is_min_k_redundantly_rigid(1, 2)
True
>>> G.is_min_k_redundantly_rigid(2, 2)
False
>>> G = Graph([[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]])
>>> G.is_k_redundantly_rigid(1, 2)
True
>>> G.is_min_k_redundantly_rigid(1, 2)
False
""" # noqa: E501
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if not isinstance(k, int):
raise TypeError(f"k needs to be a nonnegative integer, but is {k}!")
if nx.number_of_selfloops(self) > 0:
raise LoopError()
n = self.number_of_nodes()
m = self.number_of_edges()
# use bound from thm-minimal-1-edge-redundant-upper-edge-bound-dim2
if dim == 2:
if k == 1:
if n >= 7 and m > 3 * n - 9:
return False
if not self.is_k_redundantly_rigid(k, dim, combinatorial):
return False
# for the following we need to know that the graph is k-redundantly rigid
if (
dim == 2
and (
# basic edge bound
(k == 1 and m == 2 * n - 2)
or
# edge bound from :prf:ref:`thm-1-edge-redundant-edge-bound-dim2`
(k == 2 and n >= 5 and m == 2 * n)
or
# edge bound from :prf:ref:`thm-k-edge-redundant-edge-bound-dim2`
(k >= 3 and n >= 6 * (k + 1) + 23 and m == ((k + 2) * n + 1) // 2)
)
) or (
dim == 3
and (
# edge bound from :prf:ref:`thm-2-edge-redundant-edge-bound-dim3`
(k == 2 and n >= 14 and m == 3 * n - 4)
or
# edge bound from :prf:ref:`thm-k-edge-redundant-edge-bound-dim3`
(
k >= 4
and n >= 12 * (k + 1) + 10
and n % 2 == 0
and m == ((k + 3) * n + 1) // 2
)
)
):
return True
# in all other cases check by definition
G = deepcopy(self)
for edge in self.edge_list():
G.delete_edges([edge])
if G.is_k_redundantly_rigid(k, dim, combinatorial):
return False
G.add_edges([edge])
return True
[docs]
@doc_category("Generic rigidity")
def is_rigid(self, dim: int = 2, combinatorial: bool = True) -> bool:
"""
Check whether the graph is :prf:ref:`(generically) dim-rigid <def-gen-rigid>`.
Examples
--------
>>> G = Graph([(0,1), (1,2), (2,3), (3,0)])
>>> G.is_rigid()
False
>>> G.add_edge(0,2)
>>> G.is_rigid()
True
TODO
----
Pebble game algorithm for d=2.
Notes
-----
* dim=1: Connectivity
* dim=2: Pebble-game/(2,3)-rigidity
* dim>=1: Rigidity Matrix if ``combinatorial==False``
By default, the graph is in dimension two and a combinatorial check is employed.
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if not isinstance(combinatorial, bool):
raise TypeError(
"combinatorial determines the method of rigidity-computation. "
"It needs to be a Boolean."
)
if nx.number_of_selfloops(self) > 0:
raise LoopError()
elif dim == 1:
return nx.is_connected(self)
elif dim == 2 and combinatorial:
deficiency = -(2 * self.number_of_nodes() - 3) + self.number_of_edges()
if deficiency < 0:
return False
else:
self._build_pebble_digraph(2, 3)
return (
self._pebble_digraph.number_of_edges()
== 2 * self.number_of_nodes() - 3
)
elif not combinatorial:
from pyrigi.framework import Framework
F = Framework.Random(self, dim)
return F.is_inf_rigid()
else:
raise ValueError(
f"The Dimension for combinatorial computation must be either 1 or 2, "
f"but is {dim}"
)
[docs]
@doc_category("Generic rigidity")
def is_min_rigid(
self,
dim: int = 2,
combinatorial: bool = True,
use_precomputed_pebble_digraph: bool = False,
) -> bool:
"""
Check whether the graph is :prf:ref:`minimally (generically) dim-rigid
<def-min-rigid-graph>`.
Parameters
----------
use_precomputed_pebble_digraph:
Only relevant if ``dim=2`` and ``combinatorial=True``.
If ``True``, the pebble digraph present in the cache is used.
If ``False``, recompute the pebble digraph.
Use ``True`` only if you are certain that the pebble game digraph
is consistent with the graph.
Examples
--------
>>> G = Graph([(0,1), (1,2), (2,3), (3,0), (1,3)])
>>> G.is_min_rigid()
True
>>> G.add_edge(0,2)
>>> G.is_min_rigid()
False
Notes
-----
* dim=1: Tree
* dim=2: Pebble-game/(2,3)-tight
* dim>=1: Probabilistic Rigidity Matrix (maybe symbolic?)
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if not isinstance(combinatorial, bool):
raise TypeError(
"combinatorial determines the method of rigidity-computation. "
"It needs to be a Boolean."
)
if nx.number_of_selfloops(self) > 0:
raise LoopError()
elif dim == 1 and combinatorial:
return nx.is_tree(self)
elif dim == 2 and combinatorial:
return self.is_tight(
2,
3,
algorithm="pebble",
use_precomputed_pebble_digraph=use_precomputed_pebble_digraph,
)
elif not combinatorial:
from pyrigi.framework import Framework
F = Framework.Random(self, dim)
return F.is_min_inf_rigid()
else:
raise ValueError(
f"The dimension for combinatorial computation must be either 1 or 2, "
f"but is {dim}"
)
[docs]
@doc_category("Generic rigidity")
def is_globally_rigid(self, dim: int = 2) -> bool:
"""
Check whether the graph is :prf:ref:`globally dim-rigid
<def-globally-rigid-graph>`.
TODO
----
missing definition, implementation for dim>=3
Examples
--------
>>> G = Graph([(0,1), (1,2), (2,0)])
>>> G.is_globally_rigid()
True
Notes
-----
* dim=1: 2-connectivity
* dim=2: redundantly rigid+3-connected
* dim>=3: Randomized Rigidity Matrix => Stress (symbolic maybe?)
By default, the graph is in dimension 2.
A complete graph is automatically globally rigid
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if nx.number_of_selfloops(self) > 0:
raise LoopError()
elif dim == 1:
if (self.number_of_nodes() == 2 and self.number_of_edges() == 1) or (
self.number_of_nodes() == 1 or self.number_of_nodes() == 0
):
return True
return self.vertex_connectivity() >= 2
elif dim == 2:
if (
(self.number_of_nodes() == 3 and self.number_of_edges() == 3)
or (self.number_of_nodes() == 2 and self.number_of_edges() == 1)
or (self.number_of_nodes() == 1 or self.number_of_nodes() == 0)
):
return True
return self.is_redundantly_rigid() and self.vertex_connectivity() >= 3
else:
# Random sampling from [1,N] for N depending quadratically on number
# of vertices.
raise NotImplementedError()
[docs]
@doc_category("Partially implemented")
def is_Rd_dependent(
self, dim: int = 2, use_precomputed_pebble_digraph: bool = False
) -> bool:
"""
Notes
-----
* dim=1: Graphic Matroid
* dim=2: not (2,3)-sparse
* dim>=1: Compute the rank of the rigidity matrix and compare with edge count
use_precomputed_pebble_digraph:
Only relevant if ``dim=2``.
If ``True``, the pebble digraph present in the cache is used.
If ``False``, recompute the pebble digraph.
Use ``True`` only if you are certain that the pebble game digraph
is consistent with the graph.
TODO
-----
Add unit tests
"""
return not self.is_Rd_independent(
dim, use_precomputed_pebble_digraph=use_precomputed_pebble_digraph
)
[docs]
@doc_category("Partially implemented")
def is_Rd_independent(
self, dim: int = 2, use_precomputed_pebble_digraph: bool = False
) -> bool:
"""
Notes
-----
* dim=1: Graphic Matroid
* dim=2: (2,3)-sparse
* dim>=1: Compute the rank of the rigidity matrix and compare with edge count
use_precomputed_pebble_digraph:
Only relevant if ``dim=2``.
If ``True``, the pebble digraph present in the cache is used.
If ``False``, recompute the pebble digraph.
Use ``True`` only if you are certain that the pebble game digraph
is consistent with the graph.
TODO
-----
Add unit tests
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if nx.number_of_selfloops(self) > 0:
raise LoopError()
if dim == 1:
return len(self.cycle_basis()) == 0
if dim == 2:
self.is_sparse(
2, 3, use_precomputed_pebble_digraph=use_precomputed_pebble_digraph
)
raise NotImplementedError()
[docs]
@doc_category("Partially implemented")
def is_Rd_circuit(
self, dim: int = 2, use_precomputed_pebble_digraph: bool = False
) -> bool:
"""
Notes
-----
* dim=1: Graphic Matroid
* dim=2: It is not sparse, but remove any edge and it becomes sparse
Fundamental circuit is the whole graph
* Not combinatorially:
* dim>=1: Dependent + Remove every edge and compute the rigidity matrix' rank
use_precomputed_pebble_digraph:
Only relevant if ``dim=2``.
If ``True``, the pebble digraph present in the cache is used.
If ``False``, recompute the pebble digraph.
Use ``True`` only if you are certain that the pebble game digraph
is consistent with the graph.
TODO
-----
Add unit tests
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if nx.number_of_selfloops(self) > 0:
raise LoopError()
if dim == 1:
if not self.is_connected():
return False
# Check if every vertex has degree 2
for vertex in self.nodes():
if self.degree(vertex) != 2:
return False
return True
if dim == 2:
# get max sparse sugraph and check the fundamental circuit of
# the one last edge
if self.number_of_edges() != 2 * self.number_of_nodes() - 2:
return False
max_sparse_subgraph = self.spanning_sparse_subgraph(
K=2, L=3, use_precomputed_pebble_digraph=use_precomputed_pebble_digraph
)
if max_sparse_subgraph.number_of_edges() != 2 * self.number_of_nodes() - 3:
return False
remaining_edge = list(set(self.edges()) - set(max_sparse_subgraph.edges()))
if len(remaining_edge) != 1:
# this should not happen
raise RuntimeError
return self._pebble_digraph.fundamental_circuit(
u=remaining_edge[0][0], v=remaining_edge[0][1], K=2, L=3
)
raise NotImplementedError()
[docs]
@doc_category("Waiting for implementation")
def is_Rd_closed(self, dim: int = 2) -> bool:
"""
Notes
-----
* dim=1: Graphic Matroid
* dim=2: ??
* dim>=1: Adding any edge does not increase the rigidity matrix rank
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if nx.number_of_selfloops(self) > 0:
raise LoopError()
raise NotImplementedError()
[docs]
@doc_category("Generic rigidity")
def max_rigid_subgraphs(self, dim: int = 2) -> List[Graph]:
"""
List the vertex sets inducing vertex-maximal rigid subgraphs.
Definitions
-----
:prf:ref:`Maximal rigid subgraph <def-maximal-rigid-subgraph>`
TODO
----
missing definition, tests
Notes
-----
We only return nontrivial subgraphs, meaning that there need to be at
least ``dim+1`` vertices present. If the graph itself is rigid, it is clearly
maximal and is returned.
Examples
--------
>>> G = Graph([(0,1), (1,2), (2,3), (3,0)])
>>> G.max_rigid_subgraphs()
[]
>>> G = Graph([(0,1), (1,2), (2,3), (3,4), (4,5), (5,0), (0,2), (5,3)])
>>> G.is_rigid()
False
>>> G.max_rigid_subgraphs()
[[0, 1, 2], [3, 4, 5]]
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if nx.number_of_selfloops(self) > 0:
raise LoopError()
if not nx.is_connected(self):
res = []
for comp in nx.connected_components(self):
res += self.subgraph(comp).max_rigid_subgraphs(dim)
return res
if self.number_of_nodes() <= dim:
return []
if self.is_rigid(dim):
return [self]
rigid_subgraphs = {
tuple(vertex_subset): True
for r in range(dim + 1, self.number_of_nodes() - 1)
for vertex_subset in combinations(self.nodes, r)
if self.subgraph(vertex_subset).is_rigid(dim)
}
sorted_rigid_subgraphs = sorted(
rigid_subgraphs.keys(), key=lambda t: len(t), reverse=True
)
for i, H1 in enumerate(sorted_rigid_subgraphs):
if rigid_subgraphs[H1] and i + 1 < len(sorted_rigid_subgraphs):
for H2 in sorted_rigid_subgraphs[i + 1 :]:
if set(H2).issubset(set(H1)):
rigid_subgraphs[H2] = False
return [list(H) for H, is_max in rigid_subgraphs.items() if is_max]
[docs]
@doc_category("Generic rigidity")
def min_rigid_subgraphs(self, dim: int = 2) -> List[Graph]:
"""
List the vertex sets inducing vertex-minimal non-trivial rigid subgraphs.
Definitions
-----
:prf:ref:`Minimal rigid subgraph <def-minimal-rigid-subgraph>`
TODO
----
missing definition, tests
Notes
-----
We only return nontrivial subgraphs, meaning that there need to be at
least ``dim+1`` vertices present.
Examples
--------
>>> import pyrigi.graphDB as graphs
>>> G = graphs.CompleteBipartite(3, 3)
>>> G.is_rigid()
True
>>> G.min_rigid_subgraphs()
[[0, 1, 2, 3, 4, 5]]
>>> G = graphs.ThreePrism()
>>> G.is_rigid()
True
>>> G.min_rigid_subgraphs()
[[0, 1, 2], [3, 4, 5]]
"""
if not isinstance(dim, int) or dim < 1:
raise TypeError(
f"The dimension needs to be a positive integer, but is {dim}!"
)
if nx.number_of_selfloops(self) > 0:
raise LoopError()
if not nx.is_connected(self):
res = []
for comp in nx.connected_components(self):
res += self.subgraph(comp).min_rigid_subgraphs(dim)
return res
if self.number_of_nodes() <= dim:
return []
rigid_subgraphs = {
tuple(vertex_subset): True
for r in range(dim + 1, self.number_of_nodes() + 1)
for vertex_subset in combinations(self.nodes, r)
if self.subgraph(vertex_subset).is_rigid(dim)
}
sorted_rigid_subgraphs = sorted(rigid_subgraphs.keys(), key=lambda t: len(t))
for i, H1 in enumerate(sorted_rigid_subgraphs):
if rigid_subgraphs[H1] and i + 1 < len(sorted_rigid_subgraphs):
for H2 in sorted_rigid_subgraphs[i + 1 :]:
if set(H1).issubset(set(H2)):
rigid_subgraphs[H2] = False
return [list(H) for H, is_min in rigid_subgraphs.items() if is_min]
[docs]
@doc_category("General graph theoretical properties")
def is_isomorphic(self, graph: Graph) -> bool:
"""
Check whether two graphs are isomorphic.
Notes
-----
For further details, see :func:`networkx.algorithms.isomorphism.is_isomorphic`.
Examples
--------
>>> G = Graph([(0,1), (1,2)])
>>> G_ = Graph([('b','c'), ('c','a')])
>>> G.is_isomorphic(G_)
True
"""
return nx.is_isomorphic(self, graph)
[docs]
@doc_category("Other")
def to_int(self, vertex_order: List[Vertex] = None) -> int:
r"""
Return the integer representation of the graph.
The graph integer representation is the integer whose binary
expansion is given by the sequence obtained by concatenation
of the rows of the upper triangle of the adjacency matrix,
excluding the diagonal.
Parameters
----------
vertex_order:
By listing vertices in the preferred order, the adjacency matrix
is computed with the given order. If no vertex order is
provided, :meth:`~.Graph.vertex_list()` is used.
Examples
--------
>>> G = Graph([(0,1), (1,2)])
>>> G.adjacency_matrix()
Matrix([
[0, 1, 0],
[1, 0, 1],
[0, 1, 0]])
>>> G.to_int()
5
TODO
----
Implement taking canonical before computing the integer representation.
Tests.
"""
if self.number_of_edges() == 0:
raise ValueError(
"The integer representation is only defined "
"for graphs with at least one edge."
)
if self.min_degree() == 0:
raise ValueError(
"The integer representation only works "
"for graphs without isolated vertices."
)
if nx.number_of_selfloops(self) == 0:
M = self.adjacency_matrix(vertex_order)
upper_diag = [
str(b) for i, row in enumerate(M.tolist()) for b in row[i + 1 :]
]
return int("".join(upper_diag), 2)
else:
raise LoopError()
[docs]
@classmethod
@doc_category("Class methods")
def from_int(cls, N: int) -> Graph:
"""
Return a graph given its integer representation.
See :meth:`to_int` for the description
of the integer representation.
"""
if not isinstance(N, int):
raise TypeError(f"The parameter n has to be an integer, not {type(N)}.")
if N <= 0:
raise ValueError(f"The parameter n has to be positive, not {N}.")
L = bin(N)[2:]
n = math.ceil((1 + math.sqrt(1 + 8 * len(L))) / 2)
rows = []
s = 0
L = "".join(["0" for _ in range(int(n * (n - 1) / 2) - len(L))]) + L
for i in range(n):
rows.append(
[0 for _ in range(i + 1)] + [int(k) for k in L[s : s + (n - i - 1)]]
)
s += n - i - 1
adjMatrix = Matrix(rows)
return Graph.from_adjacency_matrix(adjMatrix + adjMatrix.transpose())
[docs]
@classmethod
@doc_category("Class methods")
def from_adjacency_matrix(cls, M: Matrix) -> Graph:
"""
Create a graph from a given adjacency matrix.
Examples
--------
>>> M = Matrix([[0,1],[1,0]])
>>> G = Graph.from_adjacency_matrix(M)
>>> print(G)
Graph with vertices [0, 1] and edges [[0, 1]]
"""
if not M.is_square:
raise TypeError("The matrix is not square!")
if not M.is_symmetric():
raise TypeError("The matrix is not symmetric.")
vertices = range(M.cols)
edges = []
for i, j in combinations(vertices, 2):
if not (M[i, j] == 0 or M[i, j] == 1):
raise TypeError(
"The provided adjacency matrix contains entries other than 0 and 1"
)
if M[i, j] == 1:
edges += [(i, j)]
return Graph.from_vertices_and_edges(vertices, edges)
[docs]
@doc_category("General graph theoretical properties")
def adjacency_matrix(self, vertex_order: List[Vertex] = None) -> Matrix:
"""
Return the adjacency matrix of the graph.
Parameters
----------
vertex_order:
By listing vertices in the preferred order, the adjacency matrix
can be computed in a way the user expects. If no vertex order is
provided, :meth:`~.Graph.vertex_list()` is used.
Examples
--------
>>> G = Graph([(0,1), (1,2), (1,3)])
>>> G.adjacency_matrix()
Matrix([
[0, 1, 0, 0],
[1, 0, 1, 1],
[0, 1, 0, 0],
[0, 1, 0, 0]])
Notes
-----
:func:`networkx.linalg.graphmatrix.adjacency_matrix`
requires `scipy`. To avoid unnecessary imports, the method is implemented here.
"""
if vertex_order is None:
vertex_order = self.vertex_list()
else:
if not set(self.nodes) == set(
vertex_order
) or not self.number_of_nodes() == len(vertex_order):
raise IndexError(
"The vertex_order must contain the same vertices as the graph!"
)
row_list = [
[+((v1, v2) in self.edges) for v2 in vertex_order] for v1 in vertex_order
]
return Matrix(row_list)
[docs]
@doc_category("Other")
def random_framework(self, dim: int = 2, rand_range: Union(int, List[int]) = None):
# the return type is intentionally omitted to avoid circular import
"""
Return framework with random realization.
This method calls :meth:`.Framework.Random`.
"""
from pyrigi.framework import Framework
return Framework.Random(self, dim, rand_range)
def _resolve_edge_colors(
self, edge_color: Union(str, list[list[Edge]], dict[str : list[Edge]])
) -> tuple[list, list]:
"""
Return the lists of colors and edges in the format for plotting.
"""
edge_list = self.edge_list()
edge_list_ref = []
edge_color_array = []
if isinstance(edge_color, str):
return [edge_color for _ in edge_list], edge_list
if isinstance(edge_color, list):
edges_partition = edge_color
colors = distinctipy.get_colors(
len(edges_partition), colorblind_type="Deuteranomaly", pastel_factor=0.2
)
for i, part in enumerate(edges_partition):
for e in part:
if not self.has_edge(e[0], e[1]):
raise ValueError(
"The input includes a pair that is not an edge."
)
edge_color_array.append(colors[i])
edge_list_ref.append(tuple(e))
elif isinstance(edge_color, dict):
color_edges_dict = edge_color
for color, edges in color_edges_dict.items():
for e in edges:
if not self.has_edge(e[0], e[1]):
raise ValueError(
"The input includes an edge that is not part of the framework"
)
edge_color_array.append(color)
edge_list_ref.append(tuple(e))
else:
raise ValueError("The input color_edge has none of the supported formats.")
for e in edge_list:
if (e[0], e[1]) not in edge_list_ref and (e[1], e[0]) not in edge_list_ref:
edge_color_array.append("black")
edge_list_ref.append(e)
if len(edge_list_ref) > self.number_of_edges():
multiple_colored = [
e
for e in edge_list_ref
if (edge_list_ref.count(e) > 1 or (e[1], e[0]) in edge_list_ref)
]
duplicates = []
for e in multiple_colored:
if not (e in duplicates or (e[1], e[0]) in duplicates):
duplicates.append(e)
raise ValueError(
f"The color of the edges in the following list"
f"was specified multiple times: {duplicates}."
)
return edge_color_array, edge_list_ref
[docs]
@doc_category("Other")
def layout(self, layout_type: str = "spring") -> dict[Vertex, Point]:
"""
Generate a placement of the vertices.
This method a is wrapper for the functions
:func:`~networkx.drawing.layout.spring_layout`,
:func:`~networkx.drawing.layout.random_layout`,
:func:`~networkx.drawing.layout.circular_layout`
and :func:`~networkx.drawing.layout.planar_layout`
Parameters
----------
layout_type:
The supported layouts are ``circular``, ``planar``,
``random`` and ``spring`` (default).
"""
if layout_type == "circular":
return nx.drawing.layout.circular_layout(self)
elif layout_type == "planar":
return nx.drawing.layout.planar_layout(self)
elif layout_type == "random":
return nx.drawing.layout.random_layout(self)
elif layout_type == "spring":
return nx.drawing.layout.spring_layout(self)
else:
raise ValueError(f"layout_type {layout_type} is not supported.")
[docs]
@doc_category("Other")
def plot(
self,
placement: dict[Vertex, Point] = None,
layout: str = "spring",
vertex_size: int = 300,
vertex_color: str = "#4169E1",
vertex_shape: str = "o",
vertex_labels: bool = True,
edge_width: float = 2.5,
edge_color: Union(str, list[list[Edge]], dict[str : list[Edge]]) = "black",
edge_style: str = "solid",
font_color: str = "whitesmoke",
canvas_width: float = 6.4,
canvas_height: float = 4.8,
aspect_ratio: float = 1.0,
**kwargs,
) -> None:
"""
Plot the graph.
See tutorial Plotting for illustration of the options.
Parameters
----------
placement:
If ``placement`` is not specified,
then it is generated depending on parameter ``layout``.
layout:
The possibilities are ``spring`` (default), ``circular``,
``random`` or ``planar``, see also :meth:`~Graph.layout`.
vertex_size:
The size of the vertices.
vertex_color:
The color of the vertices. The color can be a string or an rgb (or rgba)
tuple of floats from 0-1.
vertex_shape:
The shape of the vertices specified as as matplotlib.scatter
marker, one of ``so^>v<dph8``.
vertex_labels:
If ``True`` (default), vertex labels are displayed.
edge_width:
edge_color:
If a single color is given as a string or rgb (or rgba) tuple
of floats from 0-1, then all edges get this color.
If a (possibly incomplete) partition of the edges is given,
then each part gets a different color.
If a dictionary from colors to a list of edge is given,
edges are colored accordingly.
The edges missing in the partition/dictionary, are colored black.
edge_style:
Edge line style: ``-``/``solid``, ``--``/``dashed``,
``-.``/``dashdot`` or ``:``/``dotted``. By default '-'.
font_size:
The size of the font used for the labels.
font_color:
The color of the font used for the labels.
canvas_width:
The width of the canvas in inches.
canvas_height:
The height of the canvas in inches.
aspect_ratio:
The ratio of y-unit to x-unit. By default 1.0.
"""
fig, ax = plt.subplots()
ax.set_adjustable("datalim")
fig.set_figwidth(canvas_width)
fig.set_figheight(canvas_height)
ax.set_aspect(aspect_ratio)
edge_color_array, edge_list_ref = self._resolve_edge_colors(edge_color)
if placement is None:
placement = self.layout(layout)
nx.draw(
self,
pos=placement,
ax=ax,
node_size=vertex_size,
node_color=vertex_color,
node_shape=vertex_shape,
with_labels=vertex_labels,
width=edge_width,
edge_color=edge_color_array,
font_color=font_color,
edgelist=edge_list_ref,
style=edge_style,
**kwargs,
)
plt.show()
Graph.__doc__ = Graph.__doc__.replace(
"METHODS",
generate_category_tables(
Graph,
1,
[
"Attribute getters",
"Class methods",
"Graph manipulation",
"General graph theoretical properties",
"Generic rigidity",
"Sparseness",
"Other",
"Waiting for implementation",
],
include_all=False,
),
)