two version of R2R are hereHEADmaster

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diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/graphical.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/graphical.py
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+"""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",
+]
+
+
+@nx._dispatchable(graphs=None)
+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
+
+    To test a non-graphical sequence:
+    >>> sequence_list = [d for n, d in G.degree()]
+    >>> sequence_list[-1] += 1
+    >>> nx.is_graphical(sequence_list)
+    False
+
+    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
+
+
+@nx._dispatchable(graphs=None)
+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.
+
+    Examples
+    --------
+    >>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (3, 4), (4, 2), (5, 1), (5, 4)])
+    >>> sequence = (d for _, d in G.degree())
+    >>> nx.is_valid_degree_sequence_havel_hakimi(sequence)
+    True
+
+    To test a non-valid sequence:
+    >>> sequence_list = [d for _, d in G.degree()]
+    >>> sequence_list[-1] += 1
+    >>> nx.is_valid_degree_sequence_havel_hakimi(sequence_list)
+    False
+
+    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
+
+
+@nx._dispatchable(graphs=None)
+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.
+
+    Examples
+    --------
+    >>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (3, 4), (4, 2), (5, 1), (5, 4)])
+    >>> sequence = (d for _, d in G.degree())
+    >>> nx.is_valid_degree_sequence_erdos_gallai(sequence)
+    True
+
+    To test a non-valid sequence:
+    >>> sequence_list = [d for _, d in G.degree()]
+    >>> sequence_list[-1] += 1
+    >>> nx.is_valid_degree_sequence_erdos_gallai(sequence_list)
+    False
+
+    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
+
+
+@nx._dispatchable(graphs=None)
+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.
+
+    Examples
+    --------
+    >>> G = nx.MultiGraph([(1, 2), (1, 3), (2, 3), (3, 4), (4, 2), (5, 1), (5, 4)])
+    >>> sequence = (d for _, d in G.degree())
+    >>> nx.is_multigraphical(sequence)
+    True
+
+    To test a non-multigraphical sequence:
+    >>> sequence_list = [d for _, d in G.degree()]
+    >>> sequence_list[-1] += 1
+    >>> nx.is_multigraphical(sequence_list)
+    False
+
+    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
+
+
+@nx._dispatchable(graphs=None)
+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.
+
+    Examples
+    --------
+    >>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (3, 4), (4, 2), (5, 1), (5, 4)])
+    >>> sequence = (d for _, d in G.degree())
+    >>> nx.is_pseudographical(sequence)
+    True
+
+    To test a non-pseudographical sequence:
+    >>> sequence_list = [d for _, d in G.degree()]
+    >>> sequence_list[-1] += 1
+    >>> nx.is_pseudographical(sequence_list)
+    False
+
+    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
+
+
+@nx._dispatchable(graphs=None)
+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.
+
+    Examples
+    --------
+    >>> G = nx.DiGraph([(1, 2), (1, 3), (2, 3), (3, 4), (4, 2), (5, 1), (5, 4)])
+    >>> in_seq = (d for n, d in G.in_degree())
+    >>> out_seq = (d for n, d in G.out_degree())
+    >>> nx.is_digraphical(in_seq, out_seq)
+    True
+
+    To test a non-digraphical scenario:
+    >>> in_seq_list = [d for n, d in G.in_degree()]
+    >>> in_seq_list[-1] += 1
+    >>> nx.is_digraphical(in_seq_list, out_seq)
+    False
+
+    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