+ )i=XRt^RIt^RIt.R Ot]P !RR7R Rl4tRt]P !RR7R4t]P !RR7R4t ]P !RR7R4t ]P !RR7R4t ]P !RR7R 4t R#) zTest sequences for graphiness.N)graphscVR8Xd\\V44pV#VR8Xd\\V44pV#Rp\P!V4h)uReturns 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. eghhz`method` must be 'eg' or 'hh')%is_valid_degree_sequence_erdos_gallailist%is_valid_degree_sequence_havel_hakiminxNetworkXException)sequencemethodvalidmsgs&& [/var/www/html/photoedit/myenv/lib/python3.14/site-packages/networkx/algorithms/graphical.py is_graphicalrsWb~5d8nE L 45d8nE L.""3''c\PPV4p\V4p^.V,p^V^^3wr4rVVFepV^8gWq8d\PhV^8gK)\ W74\ WG4WW,V^,3wr4rVW';;,^, uu&Kg V^,'gWVV^, ,8d\PhW4WVV3#))r utilsmake_list_of_intslenNetworkXUnfeasiblemaxmin) deg_sequencepnum_degsdmaxdmindsumnds& r_basic_graphical_testsr"Ls88--l;L LAsQwHQ1*D  q5AF'' ' U"%d,D dhA"M D K1 K axx4q1u+%### t ((rc\V4wrr4pT^8Xg7^T,T,Y,^,Y,^,,8dR#^.T^,,pT^8dYQ,^8Xd T^,pKY^, 8dR#YQ,^, T^, uYQ&p^pTp\T4FSp YX,^8Xd T^,pKYX,^, T^, uYX&pT^8gK?T^, Yg&T^, pKU \T4F&p Yi,p YZ,^,T^,uYZ&pK( KR# \PdR#i;i)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. FTr"r rrange) rrrrr rmodstubsmslenkistubs & rrr`sEh(>|(L%DX AvTA$+/dkAo!FFsdQhH a%n! AID a%<%NQ.A tA+"Q%[1_a!eNHK1u"#a%  uA;D ( 2AE HNA C  sE E"!E"c\V4wrr4pT^8Xg7^T,T,Y,^,Y,^,,8dR#Rwrgr\Y^, R4Fp Y^,8dR#YZ,^8gK"YZ,p YT ,8d Y, p Y{T ,, p\T 4F;p YYl,,, pYT ,YVT ,,,, p K= Yk, pYvT^, ,Yh,, T ,8gKR# R# \PdR#i;i)u;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. FT)rrrrr$) rrrrr rr(sum_degsum_njsum_jnjdkrun_sizevs & rrrs@(>|(L%DX AvTA$+/dkAo!FF#-AD(B' A: >> 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). FT)r rr NetworkXErrorr)r rrrr!s& ris_multigraphicalr5s{Jxx11(; A$  q5Xs4|d axx4!d(?    sA**BBc\PPV4p\ T4^,^8H;'d\ T4^8# \PdR#i;i)aReturns 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). F)r rrr4sumr)r rs& ris_pseudographicalr8Hs_Pxx11(;  | q A % @ @#l*;q*@@   sA A%$A%c\PPV4p\PPV4p^^\ T4\ T43wrErg\ Yg4p^p T^8XdR#..r\ T4Fp ^^rY8d Y<,pY8d Y,,p T ^8gT^8dR#YM,Y^,\ Y4rpT ^8d$T PRT,RT ,34KwT^8gKT PRT,4K YE8wdR#\P!T 4\P!T 4R.T ^,,pT 'Eds\P!T 4wppTR,pT\ T 4\ T 4,8dR#^p\ T4FpT 'd>T 'dT ^,^,T ^,8d\P!T 4p^pM\P!T 4wppT^8XdR#T^,^8g T^8gKT^,T3TT&T^, pK \ T4FQpTT,pT^,^8d\P!T T4K3\P!T T^,4KS T^8gEKa\P!T T4EK{R# \PdR#i;i)a`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) FTr,)rr) r rrr4rrr%appendheapqheapifyheappopheappush) in_sequence out_sequencein_deg_sequenceout_deg_sequencesuminsumoutninnoutmaxnmaxinstubheapzeroheapr in_degout_degr&freeoutfreeinr'r)stuboutstubinr*s&& ris_digraphicalrQwshX((44[A8855lC !!S%93?O;PPE3 s>D E qyRh 4[Q 8&)G 7$'F A:1$~v/?UASu A: OOR'\2;7 8 q[ OOBL )  MM( MM(x519%H (!MM(3&"  CMCM1 1vAXa[^hqk-I--1$)MM($;!&!|{Q&1*#*Q;"7 uAA;DAw{x.xa1  Q; NN8W - {   s>KK65K6)rr5r8rQrr)r) __doc__r;networkxr __all__ _dispatchablerr"rrr5r8rQrrrWs$  77t)(VVrWWt//d+A+A\kkr