"""Test sequences for graphiness.
"""
import heapq
import networkx as nx
__all__ = [
"is_graphical",
"is_multigraphical",
"is_pseudographical",
"is_digraphical",
"is_valid_degree_sequence_erdos_gallai",
"is_valid_degree_sequence_havel_hakimi",
]
[docs]def is_graphical(sequence, method="eg"):
"""Returns True if sequence is a valid degree sequence.
A degree sequence is valid if some graph can realize it.
Parameters
----------
sequence : list or iterable container
A sequence of integer node degrees
method : "eg" | "hh" (default: 'eg')
The method used to validate the degree sequence.
"eg" corresponds to the Erdős-Gallai algorithm
[EG1960]_, [choudum1986]_, and
"hh" to the Havel-Hakimi algorithm
[havel1955]_, [hakimi1962]_, [CL1996]_.
Returns
-------
valid : bool
True if the sequence is a valid degree sequence and False if not.
Examples
--------
>>> G = nx.path_graph(4)
>>> sequence = (d for n, d in G.degree())
>>> nx.is_graphical(sequence)
True
References
----------
.. [EG1960] Erdős and Gallai, Mat. Lapok 11 264, 1960.
.. [choudum1986] S.A. Choudum. "A simple proof of the Erdős-Gallai theorem on
graph sequences." Bulletin of the Australian Mathematical Society, 33,
pp 67-70, 1986. https://doi.org/10.1017/S0004972700002872
.. [havel1955] Havel, V. "A Remark on the Existence of Finite Graphs"
Casopis Pest. Mat. 80, 477-480, 1955.
.. [hakimi1962] Hakimi, S. "On the Realizability of a Set of Integers as
Degrees of the Vertices of a Graph." SIAM J. Appl. Math. 10, 496-506, 1962.
.. [CL1996] G. Chartrand and L. Lesniak, "Graphs and Digraphs",
Chapman and Hall/CRC, 1996.
"""
if method == "eg":
valid = is_valid_degree_sequence_erdos_gallai(list(sequence))
elif method == "hh":
valid = is_valid_degree_sequence_havel_hakimi(list(sequence))
else:
msg = "`method` must be 'eg' or 'hh'"
raise nx.NetworkXException(msg)
return valid
def _basic_graphical_tests(deg_sequence):
# Sort and perform some simple tests on the sequence
deg_sequence = nx.utils.make_list_of_ints(deg_sequence)
p = len(deg_sequence)
num_degs = [0] * p
dmax, dmin, dsum, n = 0, p, 0, 0
for d in deg_sequence:
# Reject if degree is negative or larger than the sequence length
if d < 0 or d >= p:
raise nx.NetworkXUnfeasible
# Process only the non-zero integers
elif d > 0:
dmax, dmin, dsum, n = max(dmax, d), min(dmin, d), dsum + d, n + 1
num_degs[d] += 1
# Reject sequence if it has odd sum or is oversaturated
if dsum % 2 or dsum > n * (n - 1):
raise nx.NetworkXUnfeasible
return dmax, dmin, dsum, n, num_degs
[docs]def is_valid_degree_sequence_havel_hakimi(deg_sequence):
r"""Returns True if deg_sequence can be realized by a simple graph.
The validation proceeds using the Havel-Hakimi theorem
[havel1955]_, [hakimi1962]_, [CL1996]_.
Worst-case run time is $O(s)$ where $s$ is the sum of the sequence.
Parameters
----------
deg_sequence : list
A list of integers where each element specifies the degree of a node
in a graph.
Returns
-------
valid : bool
True if deg_sequence is graphical and False if not.
Notes
-----
The ZZ condition says that for the sequence d if
.. math::
|d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)}
then d is graphical. This was shown in Theorem 6 in [1]_.
References
----------
.. [1] I.E. Zverovich and V.E. Zverovich. "Contributions to the theory
of graphic sequences", Discrete Mathematics, 105, pp. 292-303 (1992).
.. [havel1955] Havel, V. "A Remark on the Existence of Finite Graphs"
Casopis Pest. Mat. 80, 477-480, 1955.
.. [hakimi1962] Hakimi, S. "On the Realizability of a Set of Integers as
Degrees of the Vertices of a Graph." SIAM J. Appl. Math. 10, 496-506, 1962.
.. [CL1996] G. Chartrand and L. Lesniak, "Graphs and Digraphs",
Chapman and Hall/CRC, 1996.
"""
try:
dmax, dmin, dsum, n, num_degs = _basic_graphical_tests(deg_sequence)
except nx.NetworkXUnfeasible:
return False
# Accept if sequence has no non-zero degrees or passes the ZZ condition
if n == 0 or 4 * dmin * n >= (dmax + dmin + 1) * (dmax + dmin + 1):
return True
modstubs = [0] * (dmax + 1)
# Successively reduce degree sequence by removing the maximum degree
while n > 0:
# Retrieve the maximum degree in the sequence
while num_degs[dmax] == 0:
dmax -= 1
# If there are not enough stubs to connect to, then the sequence is
# not graphical
if dmax > n - 1:
return False
# Remove largest stub in list
num_degs[dmax], n = num_degs[dmax] - 1, n - 1
# Reduce the next dmax largest stubs
mslen = 0
k = dmax
for i in range(dmax):
while num_degs[k] == 0:
k -= 1
num_degs[k], n = num_degs[k] - 1, n - 1
if k > 1:
modstubs[mslen] = k - 1
mslen += 1
# Add back to the list any non-zero stubs that were removed
for i in range(mslen):
stub = modstubs[i]
num_degs[stub], n = num_degs[stub] + 1, n + 1
return True
[docs]def is_valid_degree_sequence_erdos_gallai(deg_sequence):
r"""Returns True if deg_sequence can be realized by a simple graph.
The validation is done using the Erdős-Gallai theorem [EG1960]_.
Parameters
----------
deg_sequence : list
A list of integers
Returns
-------
valid : bool
True if deg_sequence is graphical and False if not.
Notes
-----
This implementation uses an equivalent form of the Erdős-Gallai criterion.
Worst-case run time is $O(n)$ where $n$ is the length of the sequence.
Specifically, a sequence d is graphical if and only if the
sum of the sequence is even and for all strong indices k in the sequence,
.. math::
\sum_{i=1}^{k} d_i \leq k(k-1) + \sum_{j=k+1}^{n} \min(d_i,k)
= k(n-1) - ( k \sum_{j=0}^{k-1} n_j - \sum_{j=0}^{k-1} j n_j )
A strong index k is any index where d_k >= k and the value n_j is the
number of occurrences of j in d. The maximal strong index is called the
Durfee index.
This particular rearrangement comes from the proof of Theorem 3 in [2]_.
The ZZ condition says that for the sequence d if
.. math::
|d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)}
then d is graphical. This was shown in Theorem 6 in [2]_.
References
----------
.. [1] A. Tripathi and S. Vijay. "A note on a theorem of Erdős & Gallai",
Discrete Mathematics, 265, pp. 417-420 (2003).
.. [2] I.E. Zverovich and V.E. Zverovich. "Contributions to the theory
of graphic sequences", Discrete Mathematics, 105, pp. 292-303 (1992).
.. [EG1960] Erdős and Gallai, Mat. Lapok 11 264, 1960.
"""
try:
dmax, dmin, dsum, n, num_degs = _basic_graphical_tests(deg_sequence)
except nx.NetworkXUnfeasible:
return False
# Accept if sequence has no non-zero degrees or passes the ZZ condition
if n == 0 or 4 * dmin * n >= (dmax + dmin + 1) * (dmax + dmin + 1):
return True
# Perform the EG checks using the reformulation of Zverovich and Zverovich
k, sum_deg, sum_nj, sum_jnj = 0, 0, 0, 0
for dk in range(dmax, dmin - 1, -1):
if dk < k + 1: # Check if already past Durfee index
return True
if num_degs[dk] > 0:
run_size = num_degs[dk] # Process a run of identical-valued degrees
if dk < k + run_size: # Check if end of run is past Durfee index
run_size = dk - k # Adjust back to Durfee index
sum_deg += run_size * dk
for v in range(run_size):
sum_nj += num_degs[k + v]
sum_jnj += (k + v) * num_degs[k + v]
k += run_size
if sum_deg > k * (n - 1) - k * sum_nj + sum_jnj:
return False
return True
[docs]def is_multigraphical(sequence):
"""Returns True if some multigraph can realize the sequence.
Parameters
----------
sequence : list
A list of integers
Returns
-------
valid : bool
True if deg_sequence is a multigraphic degree sequence and False if not.
Notes
-----
The worst-case run time is $O(n)$ where $n$ is the length of the sequence.
References
----------
.. [1] S. L. Hakimi. "On the realizability of a set of integers as
degrees of the vertices of a linear graph", J. SIAM, 10, pp. 496-506
(1962).
"""
try:
deg_sequence = nx.utils.make_list_of_ints(sequence)
except nx.NetworkXError:
return False
dsum, dmax = 0, 0
for d in deg_sequence:
if d < 0:
return False
dsum, dmax = dsum + d, max(dmax, d)
if dsum % 2 or dsum < 2 * dmax:
return False
return True
[docs]def is_pseudographical(sequence):
"""Returns True if some pseudograph can realize the sequence.
Every nonnegative integer sequence with an even sum is pseudographical
(see [1]_).
Parameters
----------
sequence : list or iterable container
A sequence of integer node degrees
Returns
-------
valid : bool
True if the sequence is a pseudographic degree sequence and False if not.
Notes
-----
The worst-case run time is $O(n)$ where n is the length of the sequence.
References
----------
.. [1] F. Boesch and F. Harary. "Line removal algorithms for graphs
and their degree lists", IEEE Trans. Circuits and Systems, CAS-23(12),
pp. 778-782 (1976).
"""
try:
deg_sequence = nx.utils.make_list_of_ints(sequence)
except nx.NetworkXError:
return False
return sum(deg_sequence) % 2 == 0 and min(deg_sequence) >= 0
[docs]def is_digraphical(in_sequence, out_sequence):
r"""Returns True if some directed graph can realize the in- and out-degree
sequences.
Parameters
----------
in_sequence : list or iterable container
A sequence of integer node in-degrees
out_sequence : list or iterable container
A sequence of integer node out-degrees
Returns
-------
valid : bool
True if in and out-sequences are digraphic False if not.
Notes
-----
This algorithm is from Kleitman and Wang [1]_.
The worst case runtime is $O(s \times \log n)$ where $s$ and $n$ are the
sum and length of the sequences respectively.
References
----------
.. [1] D.J. Kleitman and D.L. Wang
Algorithms for Constructing Graphs and Digraphs with Given Valences
and Factors, Discrete Mathematics, 6(1), pp. 79-88 (1973)
"""
try:
in_deg_sequence = nx.utils.make_list_of_ints(in_sequence)
out_deg_sequence = nx.utils.make_list_of_ints(out_sequence)
except nx.NetworkXError:
return False
# Process the sequences and form two heaps to store degree pairs with
# either zero or non-zero out degrees
sumin, sumout, nin, nout = 0, 0, len(in_deg_sequence), len(out_deg_sequence)
maxn = max(nin, nout)
maxin = 0
if maxn == 0:
return True
stubheap, zeroheap = [], []
for n in range(maxn):
in_deg, out_deg = 0, 0
if n < nout:
out_deg = out_deg_sequence[n]
if n < nin:
in_deg = in_deg_sequence[n]
if in_deg < 0 or out_deg < 0:
return False
sumin, sumout, maxin = sumin + in_deg, sumout + out_deg, max(maxin, in_deg)
if in_deg > 0:
stubheap.append((-1 * out_deg, -1 * in_deg))
elif out_deg > 0:
zeroheap.append(-1 * out_deg)
if sumin != sumout:
return False
heapq.heapify(stubheap)
heapq.heapify(zeroheap)
modstubs = [(0, 0)] * (maxin + 1)
# Successively reduce degree sequence by removing the maximum out degree
while stubheap:
# Take the first value in the sequence with non-zero in degree
(freeout, freein) = heapq.heappop(stubheap)
freein *= -1
if freein > len(stubheap) + len(zeroheap):
return False
# Attach out stubs to the nodes with the most in stubs
mslen = 0
for i in range(freein):
if zeroheap and (not stubheap or stubheap[0][0] > zeroheap[0]):
stubout = heapq.heappop(zeroheap)
stubin = 0
else:
(stubout, stubin) = heapq.heappop(stubheap)
if stubout == 0:
return False
# Check if target is now totally connected
if stubout + 1 < 0 or stubin < 0:
modstubs[mslen] = (stubout + 1, stubin)
mslen += 1
# Add back the nodes to the heap that still have available stubs
for i in range(mslen):
stub = modstubs[i]
if stub[1] < 0:
heapq.heappush(stubheap, stub)
else:
heapq.heappush(zeroheap, stub[0])
if freeout < 0:
heapq.heappush(zeroheap, freeout)
return True