btj=bdZddlZddlZgdZejdd dZdZejddZejdd Z ejdd Z ejdd Z ejdd Z dS)zTest sequences for graphiness.N) is_graphicalis_multigraphicalis_pseudographicalis_digraphical%is_valid_degree_sequence_erdos_gallai%is_valid_degree_sequence_havel_hakimi)graphsegc|dkrtt|}n9|dkrtt|}nd}tj||S)usReturns 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. r hhz`method` must be 'eg' or 'hh')rlistrnxNetworkXException)sequencemethodvalidmsgs [/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/networkx/algorithms/graphical.pyrrs\b~~5d8nnEE 45d8nnEE-"3''' Lctj|}t|}dg|z}d|ddf\}}}}|D]]}|dks||kr tj|dkr=t ||t ||||z|dzf\}}}}||xxdz cc<^|dzs |||dz zkr tj|||||fS)Nr)rutilsmake_list_of_intslenNetworkXUnfeasiblemaxmin) deg_sequencepnum_degsdmaxdmindsumnds r_basic_graphical_testsr'Ls8--l;;L LAsQwHQ1*D$a  q55AFF' ' UU"%dA,,D! dQhA"M D$a QKKK1 KKK ax$4!q1u+%%## tQ ((rcX t|\}}}}}n#tj$rYdSwxYw|dksd|z|z||zdz||zdzzkrdSdg|dzz}|dkr||dkr|dz}||dk||dz krdS||dz |dz c||<}d}|}t|D]F} ||dkr|dz}||dk||dz |dz c||<}|dkr |dz ||<|dz }Gt|D]} || } || dz|dzc|| <}|dkdS)aReturns 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. FrrTr'rrrange) rr"r#r$r%r!modstubsmslenkistubs rrr`sh(>|(L(L%dD!XX uu AvvTA$+/dTkAo!FFFtsdQhH a%%tn!! AIDtn!! !a%<<5%TNQ.A t  A1+""Q1+""%a[1_a!eNHQK1uu"#a% u : :AA;D ( 2AE HTNAA1 a%%2 4 ++c t|\}}}}}n#tj$rYdSwxYw|dksd|z|z||zdz||zdzzkrdSd\}}}} t||dz dD]} | |dzkrdS|| dkro|| } | || zkr| |z } || | zz }t| D]$} |||| zz }| || z||| zzz } %|| z }|||dz z||zz | zkrdSdS)uReturns 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. Frr)rT)rrrrr*) rr"r#r$r%r!r.sum_degsum_njsum_jnjdkrun_sizevs rrrsz@(>|(L(L%dD!XX uu AvvTA$+/dTkAo!FFFt#-AwD$(B''   A::44 B>> 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). FrrrrT)rrr NetworkXErrorr)rrr$r"r&s rrrsJx11(;; uuJD$ ,, q5555AXs4||d ax4!d(??u 4 "55c tj|}n#tj$rYdSwxYwt |dzdkot |dkS)a: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). Frr)rrrr<sumr)rrs rrrHsnPx11(;; uu |  q A % @#l*;*;q*@@r=cb tj|}tj|}n#tj$rYdSwxYwddt |t |f\}}}}t ||}d} |dkrdSgg} } t |D]} d\} }| |kr|| }| |kr|| } | dks|dkrdS|| z||zt | | } }}| dkr| d|zd| zfp|dkr| d|z||krdStj | tj | dg| dzz}| r;tj | \}}|dz}|t | t | zkrdSd}t |D]s}| r1| r| dd| dkrtj | }d}ntj | \}}|dkrdS|dzdks|dkr|dz|f||<|dz }tt |D]G}||}|ddkrtj | |,tj | |dH|dkrtj | || ;dS)aReturns 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) FrTr;r3r) rrrr<rrr+appendheapqheapifyheappopheappush) in_sequence out_sequencein_deg_sequenceout_deg_sequencesuminsumoutninnoutmaxnmaxinstubheapzeroheapr%in_degout_degr,freeoutfreeinr-r/stuboutstubinr0s rrrws,X(44[AA855lCC uu !!S%9%93?O;P;PPE63 sD>>D E qyytRhH 4[[ * * t88&q)G s77$Q'F A::155$v~v/?UFASASuv A:: OOR'\2;7 8 8 8 8 q[[ OOBL ) ) ) u M( M(x519%H .!M(33&"  CMMCMM1 1 15v  A < 4s>AAA)r ) __doc__rBnetworkxr__all__ _dispatchablerr'rrrrrrrr]sj$$    7777t)))(VVVrWWWt///d+A+A+A\kkkkkr