\MhD.dZddlmZddlmZddlmZddlZddl m Z ddl m Z dd gZ ejd dd Zdd ZddZe de dejd dZdZdZdZddZdS)z%Functions for generating line graphs.) defaultdict)partial) combinationsN)arbitrary_element)not_implemented_for line_graphinverse_line_graphT) returns_graphcv|rt||}nt|d|}|S)aReturns the line graph of the graph or digraph `G`. The line graph of a graph `G` has a node for each edge in `G` and an edge joining those nodes if the two edges in `G` share a common node. For directed graphs, nodes are adjacent exactly when the edges they represent form a directed path of length two. The nodes of the line graph are 2-tuples of nodes in the original graph (or 3-tuples for multigraphs, with the key of the edge as the third element). For information about self-loops and more discussion, see the **Notes** section below. Parameters ---------- G : graph A NetworkX Graph, DiGraph, MultiGraph, or MultiDigraph. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- L : graph The line graph of G. Examples -------- >>> G = nx.star_graph(3) >>> L = nx.line_graph(G) >>> print(sorted(map(sorted, L.edges()))) # makes a 3-clique, K3 [[(0, 1), (0, 2)], [(0, 1), (0, 3)], [(0, 2), (0, 3)]] Edge attributes from `G` are not copied over as node attributes in `L`, but attributes can be copied manually: >>> G = nx.path_graph(4) >>> G.add_edges_from((u, v, {"tot": u + v}) for u, v in G.edges) >>> G.edges(data=True) EdgeDataView([(0, 1, {'tot': 1}), (1, 2, {'tot': 3}), (2, 3, {'tot': 5})]) >>> H = nx.line_graph(G) >>> H.add_nodes_from((node, G.edges[node]) for node in H) >>> H.nodes(data=True) NodeDataView({(0, 1): {'tot': 1}, (2, 3): {'tot': 5}, (1, 2): {'tot': 3}}) Notes ----- Graph, node, and edge data are not propagated to the new graph. For undirected graphs, the nodes in G must be sortable, otherwise the constructed line graph may not be correct. *Self-loops in undirected graphs* For an undirected graph `G` without multiple edges, each edge can be written as a set `\{u, v\}`. Its line graph `L` has the edges of `G` as its nodes. If `x` and `y` are two nodes in `L`, then `\{x, y\}` is an edge in `L` if and only if the intersection of `x` and `y` is nonempty. Thus, the set of all edges is determined by the set of all pairwise intersections of edges in `G`. Trivially, every edge in G would have a nonzero intersection with itself, and so every node in `L` should have a self-loop. This is not so interesting, and the original context of line graphs was with simple graphs, which had no self-loops or multiple edges. The line graph was also meant to be a simple graph and thus, self-loops in `L` are not part of the standard definition of a line graph. In a pairwise intersection matrix, this is analogous to excluding the diagonal entries from the line graph definition. Self-loops and multiple edges in `G` add nodes to `L` in a natural way, and do not require any fundamental changes to the definition. It might be argued that the self-loops we excluded before should now be included. However, the self-loops are still "trivial" in some sense and thus, are usually excluded. *Self-loops in directed graphs* For a directed graph `G` without multiple edges, each edge can be written as a tuple `(u, v)`. Its line graph `L` has the edges of `G` as its nodes. If `x` and `y` are two nodes in `L`, then `(x, y)` is an edge in `L` if and only if the tail of `x` matches the head of `y`, for example, if `x = (a, b)` and `y = (b, c)` for some vertices `a`, `b`, and `c` in `G`. Due to the directed nature of the edges, it is no longer the case that every edge in `G` should have a self-loop in `L`. Now, the only time self-loops arise is if a node in `G` itself has a self-loop. So such self-loops are no longer "trivial" but instead, represent essential features of the topology of `G`. For this reason, the historical development of line digraphs is such that self-loops are included. When the graph `G` has multiple edges, once again only superficial changes are required to the definition. References ---------- * Harary, Frank, and Norman, Robert Z., "Some properties of line digraphs", Rend. Circ. Mat. Palermo, II. Ser. 9 (1960), 161--168. * Hemminger, R. L.; Beineke, L. W. (1978), "Line graphs and line digraphs", in Beineke, L. W.; Wilson, R. J., Selected Topics in Graph Theory, Academic Press Inc., pp. 271--305. ) create_usingF) selfloopsr ) is_directed _lg_directed_lg_undirected)Gr Ls X/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/networkx/generators/line.pyrrsCL }}J  6 6 6 1L I I I Hc8tjd||j}|rt |jdn|j}|D]A}||||dD]}|||B|S)a6Returns the line graph L of the (multi)digraph G. Edges in G appear as nodes in L, represented as tuples of the form (u,v) or (u,v,key) if G is a multidigraph. A node in L corresponding to the edge (u,v) is connected to every node corresponding to an edge (v,w). Parameters ---------- G : digraph A directed graph or directed multigraph. create_using : NetworkX graph constructor, optional Graph type to create. If graph instance, then cleared before populated. Default is to use the same graph class as `G`. rdefaultTkeys)nx empty_graph __class__ is_multigraphredgesadd_nodeadd_edge)rr r get_edges from_nodeto_nodes rrr{s q, <<z"_lg_undirected..s00041a!Q000rc<|d|dfS)Nrrr')edge node_indexs redge_key_functionz)_lg_undirected..edge_key_functions $q'"JtAw$777rc |g|]8}tt|ddj|ddz9S)Nkey)tuplesortedget)r(xr.s r z"_lg_undirected..sEXXXavae88899AabbEAXXXrcPg|]"}tt|f#S)r2)r4r5)r(bar/s rr8z"_lg_undirected..sC&!Q->???@@rN) rrrrrr enumeratesetlenr updateadd_edges_from) rr r rr"shiftrunodesr)r;r/r.s @@@rrrs0 q, <<>> K5 = nx.complete_graph(5) >>> P4 = nx.Graph([("a", "b"), ("b", "c"), ("c", "d")]) >>> G = nx.union(K5, P4) >>> root_graphs = [] >>> for comp in nx.connected_components(G): ... root_graphs.append(nx.inverse_line_graph(G.subgraph(comp))) >>> len(root_graphs) 2 References ---------- .. [1] Roussopoulos, N.D. , "A max {m, n} algorithm for determining the graph H from its line graph G", Information Processing Letters 2, (1973), 108--112, ISSN 0020-0190, `DOI link `_ rrzninverse_line_graph() doesn't work on an edgeless graph. Please use this function on each component separately.zA line graph as generated by NetworkX has no selfloops, so G has no inverse line graph. Please remove the selfloops from G and try again.r1zEG is not a line graph (vertex found in more than two partition cells)c36K|]}|dk|fVdS)rNr')r(rBP_counts r z%inverse_line_graph..0s- 7 7qwqzQqd 7 7rc3 K|]}|vV dSNr')r(a_bitr:s rrIz%inverse_line_graph..5s'))euz))))))r)number_of_nodesrrrGraphnumber_of_edges NetworkXErrornumber_of_selfloops_select_starting_cell_find_partitiondictfromkeysrCmaxvaluesr4add_nodes_fromranyr!) rvr;Hmsg starting_cellPprBWrHr:s @@rr r sPj a~a       ! ! a  F F Hq!fX      q Q%6%6%8%8A%=%= E s### a  A%% T s###)!,,M=))AmmAGQ''G   A AJJJ!OJJJJ  7>>  q  Us### 7 7 7 7G 7 7 777A  AQQQWa((1 ))))q))) ) )  JJq!    Hrc|\}}||vrtjd|d|||vrtjd|d|dg}||D]$}|||vr||||f%|S)z.Return list of all triangles containing edge eVertex not in graphEdge (, ) not in graph)rrPappend)rerBrZ triangle_listr7s r _trianglesrj:s DAqzz9999:::!}}>>>Q>>>???M qT,, !99  !Q + + + rc|D]0}||vrtjd|d1tt |dD]?}|d||dvr'tjd|dd|dd@t t |D]!}||D]}||vr|xxdz cc<"tfd DS) aTest whether T is an odd triangle in G Parameters ---------- G : NetworkX Graph T : 3-tuple of vertices forming triangle in G Returns ------- True is T is an odd triangle False otherwise Raises ------ NetworkXError T is not a triangle in G Notes ----- An odd triangle is one in which there exists another vertex in G which is adjacent to either exactly one or exactly all three of the vertices in the triangle. rbrcr1rrrdrerfc3,K|]}|dvVdS))rNr')r(rZT_nbrss rrIz _odd_triangle..ms,33qvayF"333333r)rCrrPlistrrintrY)rTrBrhtrZrns @r _odd_trianglersHs/2?? AGGII  "#=Q#=#=#=>> >  ,q!$$ % %JJ Q4q1w  "#HAaD#H#HAaD#H#H#HII I   F 1  Azzq Q   3333F333 3 33rc|}|g}|tt|dt|}|dkr|}t ||}|dkr|gt||z}|D]-}|D](}||kr |||vrd} tj| ).| t||tt|d||z }|dk|S)aiFind a partition of the vertices of G into cells of complete graphs Parameters ---------- G : NetworkX Graph starting_cell : tuple of vertices in G which form a cell Returns ------- List of tuples of vertices of G Raises ------ NetworkXError If a cell is not a complete subgraph then G is not a line graph r1rz>G is not a line graph (partition cell not a complete subgraph)) copyremove_edges_fromrorrOpopr>rrPrgr4) rr] G_partitionr^partitioned_verticesrBdeg_unew_cellrZr\s rrSrSpsl"&&((K A!!$|M1'E'E"F"FGGG ..  % % ' '! + + $ $ & &KN## A::sT+a.111H 4 4!44AQQk!n%<%<G!.s333 4 HHU8__ % % %  ) )$|Ha/H/H*I*I J J J H , '  % % ' '! + +( HrcT|"t|}n{|}|d|vrtjd|dd|d||dvr)d|dd|dd}tj|t ||}t |}|dkr|}n_|dkr|d}|\}} } t t ||| f} t t || | f} | dkr| dkr|}nt|| | f St||| f Sd} g}|D],}t||r| dz } | |-|d kr | dkr|}n|dz | cxkr|krpnnmt}|D]}|D]}| ||D]-}|D](}||kr |||vrd }tj|).t|}nd }tj||S) a_Select a cell to initiate _find_partition Parameters ---------- G : NetworkX Graph starting_edge: an edge to build the starting cell from Returns ------- Tuple of vertices in G Raises ------ NetworkXError If it is determined that G is not a line graph Notes ----- If starting edge not specified then pick an arbitrary edge - doesn't matter which. However, this function may call itself requiring a specific starting edge. Note that the r, s notation for counting triangles is the same as in the Roussopoulos paper cited above. Nrrbrcrzstarting_edge (rez) is not in the Graph) starting_edger1zCG is not a line graph (odd triangles do not form complete subgraph)zNG is not a line graph (incorrect number of odd triangles around starting edge)) rrrCrrPrjr>rRrsrgr=addr4)rr}rhr\ e_trianglesrr]rqr;r:cac_edgesbc_edgess odd_trianglestriangle_nodesr7rBrZs rrRrRs0 aggii ( (  Q4qwwyy "#@QqT#@#@#@AA A Q4q1w  GAaDGGAaDGGGC"3'' 'Q""K KAAvv a N1az!aV,,--z!aV,,-- q==1}} ! ,Qq!fEEEE(1a&AAA A   ( (AQ"" (Q$$Q''' 66a1ffMM Ua____1_____ UUN" * ***A"&&q))))*$ 4 4'44AAvv1AaD===!.s333 4".11MM6 "3'' ' rrK)FN)__doc__ collectionsr functoolsr itertoolsrnetworkxrnetworkx.utilsrnetworkx.utils.decoratorsr__all__ _dispatchablerrrr rjrsrSrRr'rrrs++######"""""",,,,,,999999 - .%%%i i i &%i X    <> > > > BZ  \""%%%Z Z &%#"! Z z   %4%4%4P* * * ZXXXXXXr