wg=TdZddlZddlZgdZej dd dZdZej ddZej ddZ ej dd Z ej dd Z ej dd Z y) zTest sequences for graphiness.N) is_graphicalis_multigraphicalis_pseudographicalis_digraphical%is_valid_degree_sequence_erdos_gallai%is_valid_degree_sequence_havel_hakimi)graphsc|dk(rtt|}|S|dk(rtt|}|Sd}tj|)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. eghhz`method` must be 'eg' or 'hh')rlistrnxNetworkXException)sequencemethodvalidmsgs b/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/networkx/algorithms/graphical.pyrrsWb~5d8nE L 45d8nE L.""3''cxtjj|}t|}dg|z}d|ddf\}}}}|D]T}|dks||k\rtj|dkDs#t ||t ||||z|dzf\}}}}||xxdz cc<V|dzs |||dz zkDrtj|||||fS)Nr)rutilsmake_list_of_intslenNetworkXUnfeasiblemaxmin) deg_sequencepnum_degsdmaxdmindsumnds r_basic_graphical_testsr'Ls88--l;L LAsQwHQ1*D$a  q5AF'' ' U"%dA,D! dQhA"M D$a QK1 K ax4!q1u+%### tQ ((rc  t|\}}}}}|dk(sd|z|z||zdz||zdzzk\rydg|dzz}|dkDr||dk(r|dz}||dk(r||dz kDry||dz |dz c||<}d}|}t|D]<} ||dk(r|dz}||dk(r||dz |dz c||<}|dkDs0|dz ||<|dz }>t|D]} || } || dz|dzc|| <}|dkDry#tj$rYywxYw)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$r%r!modstubsmslenkistubs rrr`sh(>|(L%dD!X AvTA$+/dTkAo!FFsdQhH a%tn! AIDtn! !a%<%TNQ.A t A1+"Q1+"%a[1_a!eNHQK1u"#a%   u :AA;D ( 2AE HTNA :- a%2 C  sC,,DDc t|\}}}}}|dk(sd|z|z||zdz||zdzzk\ryd\}}}} t||dz dD]v} | |dzkry|| dkDs|| } | || zkr| |z } || | zz }t| D]} |||| zz }| || z||| zzz } || z }|||dz z||zz | zkDsvyy#tj$rYywxYw)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$r%r!r.sum_degsum_njsum_jnjdkrun_sizevs rrrsJ@(>|(L%dD!X AvTA$+/dTkAo!FF#-AwD$(B'  A: 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$r"r&s rrrsJxx11(; JD$ , q5AXs4|d, ax4!d(?    sAA*)A*c tjj|}t |dzdk(xrt |dk\S#tj$rYywxYw)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 rrrHs\Pxx11(;  | q A % @#l*;q*@@   sAAAc tjj|}tjj|}ddt |t |f\}}}}t ||}d} |dk(rygg} } t |D]v} d\} }| |kr|| }| |kr|| } | dks|dkry|| z||zt | | } }}| dkDr| jd|zd| zf]|dkDsc| jd|zx||k7rytj| tj| dg| dzz}| rtj| \}}|dz}|t | t | zkDryd}t |D]h}| r(| r| dd| dkDrtj| }d}ntj| \}}|dk(ry|dzdks|dksZ|dz|f||<|dz }jt |D]?}||}|ddkrtj| |'tj| |dA|dkrtj| || ry#tj$rYywxYw)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:r2r) rrrr;rrr+appendheapqheapifyheappopheappush) in_sequence out_sequencein_deg_sequenceout_deg_sequencesuminsumoutninnoutmaxnmaxinstubheapzeroheapr%in_degout_degr,freeoutfreeinr-r/stuboutstubinr0s rrrwsX((44[A8855lC !!S%93?O;PPE63 sD>D E qyRhH 4[ * t8&q)G s7$Q'F A:1$v~v/?UFASuv A: OOR'\2;7 8 q[ OOBL ) *  MM( MM(x519%H !MM(3&"  CMCM1 1v AXa[^hqk-I--1$)MM($;!&!|{Q&1*#*Q;"7  u 2AA;DAw{x.xa1  2 Q; NN8W -= > {   s>II('I()r ) __doc__r@networkxr__all__ _dispatchablerr'rrrrrrrr[s$  77t)(VVrWWt//d+A+A\kkr