h=2SrSSKrSSKr/SQr\R "SS9S Sj5rSr\R "SS9S5r\R "SS9S5r \R "SS9S 5r \R "SS9S 5r \R "SS9S 5r g) zTest sequences for graphiness.N) is_graphicalis_multigraphicalis_pseudographicalis_digraphical%is_valid_degree_sequence_erdos_gallai%is_valid_degree_sequence_havel_hakimi)graphscUS:Xa[[U55nU$US:Xa[[U55nU$Sn[R"U5e)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')rlistrnxNetworkXException)sequencemethodvalidmsgs O/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/graphical.pyrrsWb~5d8nE L 45d8nE L.""3''cx[RRU5n[U5nS/U-nSUSS4up4pVUHRnUS:dXq:a[ReUS:dM&[ X75[ XG5XW-US-4up4pVX'==S- ss'MT US-(d XVUS- -:a[ReX4XVU4$)Nr)rutilsmake_list_of_intslenNetworkXUnfeasiblemaxmin) deg_sequencepnum_degsdmaxdmindsumnds r_basic_graphical_testsr'Ls88--l;L LAsQwHQ1*D  q5AF'' ' U"%d,D dhA"M D K1 K ax4q1u+%### t ((rc[U5upp4nUS:XdSU-U-X-S-X-S--:agS/US--nUS:aXQS:XaUS-nXQS:XaMXS- :agXQS- US- sXQ'nSnUn[U5H=n XXS:XaUS-nXXS:XaMXXS- US- sXX'nUS:dM1US- Xg'US- nM? [U5Hn Xin XZS-US-sXZ'nM US:aMg![Ra gf=f)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$r%r!modstubsmslenkistubs rrr`sfh(>|(L%DX AvTA$+/dkAo!FFsdQhH a%n! AIDn! a%<%NQ.A tA+"Q+"%[1_a!eNHK1u"#a%  uA;D ( 2AE HNA- a%2 C  sC--DDc[U5upp4nUS:XdSU-U-X-S-X-S--:agSupgp[XS- S5Hqn XS-:a gXZS:dMXZn XU -:aX- n X{U -- n[U 5Hn XXl-- nXU -XVU --- n M Xk- nXvUS- -Xh-- U -:dMq g g![Ra gf=f)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. Frr)rT)rrrrr*) rr"r#r$r%r!r.sum_degsum_njsum_jnjdkrun_sizevs rrrs@(>|(L%DX AvTA$+/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). FrrrrT)rrr NetworkXErrorr)rrr$r"r&s rrrsyJxx11(; JD  q5Xs4|d ax4!d(?    sAA0/A0c[RRU5n[ U5S-S:H=(a [ U5S:$![Ra gf=f)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). Frr)rrrr;sumr)rrs rrrHs\Pxx11(;  | q A % @#l*;q*@@   sAAAc([RRU5n[RRU5nSS[ U5[ U54upEpg[ Xg5nSn US:Xag//p[ U5Hvn SupX:aX<nX:aX,n U S:dUS:a gXM-X^-[ X5pnU S:aU RSU-SU -45 MZUS:dMbU RSU-5 Mx XE:wag[R"U 5 [R"U 5 S/U S--nU (Ga?[R"U 5unnUS-nU[ U 5[ U 5-:agSn[ U5HznU (a/U (aU SSU S:a[R"U 5nSnO[R"U 5unnUS:Xa gUS-S:dUS:dMkUS-U4UU'US- nM| [ U5HDnUUnUSS:a[R"U U5 M*[R"U US5 MF US:a[R"U U5 U (aGM?g![Ra gf=f)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) FrTr:r2r) rrrr;rrr+appendheapqheapifyheappopheappush) in_sequence out_sequencein_deg_sequenceout_deg_sequencesuminsumoutninnoutmaxnmaxinstubheapzeroheapr%in_degout_degr,freeoutfreeinr-r/stuboutstubinr0s rrrwsoX((44[A8855lC !!S%93?O;PPE3 s>D E qyRh 4[ 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>I::JJ)r ) __doc__r@networkxr__all__ _dispatchablerr'rrrrrrrr[s$  77t)(VVrWWt//d+A+A\kkr