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