h{D SrSSKJr SSKJr SSKJr SSKrSSK J r SSK J r SS /r \R"S S 9SS j5rSS jrSSjr\ "S5\ "S5\R"S S 9S555rSrSrSrSSjrg)z%Functions for generating line graphs.) defaultdict)partial) combinationsN)arbitrary_element)not_implemented_for line_graphinverse_line_graphT) returns_graphc\UR5(a [XS9nU$[USUS9nU$)abReturns 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 J/var/www/html/env/lib/python3.13/site-packages/networkx/generators/line.pyrrs6L }}  6 H 1L I Hc2[R"SXRS9nUR5(a[ UR SS9O UR nU"5H7nUR U5 U"US5HnURXE5 M M9 U$)a Returns the line graph L of the (multi)digraph G. Edges in G appear as nodes in L, represented as tuples of the form (u,v) or (u,v,key) if G is a multidigraph. A node in L corresponding to the edge (u,v) is connected to every node corresponding to an edge (v,w). Parameters ---------- G : digraph A directed graph or directed multigraph. create_using : NetworkX graph constructor, optional Graph type to create. If graph instance, then cleared before populated. Default is to use the same graph class as `G`. rdefaultTkeys)nx empty_graph __class__ is_multigraphredgesadd_nodeadd_edge)rr r get_edges from_nodeto_nodes rrr{sz q, upmU RXU-SVs/sHn[[X4US 95PM sn5 M@ M URU 5 U$s snnfs sn fs snf) aReturns the line graph L of the (multi)graph G. Edges in G appear as nodes in L, represented as sorted tuples of the form (u,v), or (u,v,key) if G is a multigraph. A node in L corresponding to the edge {u,v} is connected to every node corresponding to an edge that involves u or v. Parameters ---------- G : graph An undirected graph or multigraph. selfloops : bool If `True`, then self-loops are included in the line graph. If `False`, they are excluded. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Notes ----- The standard algorithm for line graphs of undirected graphs does not produce self-loops. rrTrrc$>TUSTUS4$)Nrr)edge node_indexs r _lg_undirected..sja&9:d1g;N%OrN)key)rrrrrr enumeratesettuplesortedgetlenr updateadd_edges_from)rr r rr"shiftinedge_key_functionruxnodesabr)s @rrrsR0 q, ?@/ %*U H91Ys/E 0EE$ directed multigraphc^ ^ UR5S:Xa[R"S5$UR5S:Xa.[U5nUS4nUS4m [R"UT 4/5nU$UR5S:a,UR 5S:XaSn[R "U5e[R"U5S:waSn[R "U5e[U5n[X5nURVs0sHowS_M snm UHnUHnT U==S- ss'M M [T R55S:aSn[R "U5e[U 4SjT 55n [R"5nURU5 URU 5 [URS5H4unm [!U 4SjU55(dM"UR#UT 5 M6 U$s snf) aReturns the inverse line graph of graph G. If H is a graph, and G is the line graph of H, such that G = L(H). Then H is the inverse line graph of G. Not all graphs are line graphs and these do not have an inverse line graph. In these cases this function raises a NetworkXError. Parameters ---------- G : graph A NetworkX Graph Returns ------- H : graph The inverse line graph of G. Raises ------ NetworkXNotImplemented If G is directed or a multigraph NetworkXError If G is not a line graph Notes ----- This is an implementation of the Roussopoulos algorithm[1]_. If G consists of multiple components, then the algorithm doesn't work. You should invert every component separately: >>> K5 = nx.complete_graph(5) >>> P4 = nx.Graph([("a", "b"), ("b", "c"), ("c", "d")]) >>> G = nx.union(K5, P4) >>> root_graphs = [] >>> for comp in nx.connected_components(G): ... root_graphs.append(nx.inverse_line_graph(G.subgraph(comp))) >>> len(root_graphs) 2 References ---------- .. [1] Roussopoulos, N.D. , "A max {m, n} algorithm for determining the graph H from its line graph G", Information Processing Letters 2, (1973), 108--112, ISSN 0020-0190, `DOI link `_ rrzninverse_line_graph() doesn't work on an edgeless graph. Please use this function on each component separately.zA line graph as generated by NetworkX has no selfloops, so G has no inverse line graph. Please remove the selfloops from G and try again.r,zEG is not a line graph (vertex found in more than two partition cells)c3@># UHnTUS:XdMU4v M g7f)rNr').0r:P_counts r %inverse_line_graph../s 7GqwqzQdqdGs  c3,># UH oT;v M g7fNr')rCa_bitr>s rrErF4s)qezqs)number_of_nodesrrrGraphnumber_of_edges NetworkXErrornumber_of_selfloops_select_starting_cell_find_partitionr<maxvaluesr0add_nodes_fromranyr!) rvr=Hmsg starting_cellPr:pWrDr>s @@rr r sj a~~a     ! a  F F HHq!fX    q Q%6%6%8A%= E s## a A% T s##)!,M)AWW%W!tW%G A AJ!OJ 7>> q Us## 7G 77A  AQQQWWa(1 )q) ) ) JJq! ) H&s HcUup#X ;a[R"SUS35eX0U;a[R"SUSUS35e/nXH nXPU;dM URX#U45 M" U$)z.Return list of all triangles containing edge eVertex not in graphEdge (, ) not in graph)rrMappend)rer:rU triangle_listr;s r _trianglesre9s DAz=9::!}s"QC~>??M T !9  ! + rc^UH0nX R5;dM[R"SUS35e [[ US55H4nUSXS;dM[R"SUSSUSS35e [ [ 5mUH"nXHnXQ;dM TU==S- ss'M M$ [U4S jT55$) aTest whether T is an odd triangle in G Parameters ---------- G : NetworkX Graph T : 3-tuple of vertices forming triangle in G Returns ------- True is T is an odd triangle False otherwise Raises ------ NetworkXError T is not a triangle in G Notes ----- An odd triangle is one in which there exists another vertex in G which is adjacent to either exactly one or exactly all three of the vertices in the triangle. r]r^r,rrr_r`rac34># UH nTUS;v M g7f))rNr')rCrUT_nbrss rrE _odd_triangle..ls3FqvayF"Fs)r<rrMlistrrintrT)rTr:rctrUris @r _odd_triangleroGs2 GGI ""WQC}#=> >,q!$ % Q4q1w ""VAaD6AaD6#HI I& F Azq Q  3F3 33rc<UR5nU/nUR[[US555 [U5nUR 5S:aUR 5n[ X%5nUS:waU/[X%5-nUH3nUH*nXX:wdM XU;dMSn [R"U 5e M5 UR[U55 UR[[US555 XG- nUR 5S:aMU$)a9Find a partition of the vertices of G into cells of complete graphs Parameters ---------- G : NetworkX Graph starting_cell : tuple of vertices in G which form a cell Returns ------- List of tuples of vertices of G Raises ------ NetworkXError If a cell is not a complete subgraph then G is not a line graph r,rz>G is not a line graph (partition cell not a complete subgraph)) copyremove_edges_fromrkrrLpopr3rrMrbr0) rrX G_partitionrYpartitioned_verticesr:deg_unew_cellrUrWs rrPrPos"&&(K A!!$|M1'E"FG .  % % '! + $ $ &KN# A:sT+.11H!AQ!n%<G!..s33 " HHU8_ %  ) )$|Ha/H*I J , '  % % '! +( HrcUc[UR55nOiUnUSUR5;a[R"SUSS35eUSXS;a%SUSSUSS3n[R"U5e[ X5n[ U5nUS:XaUnU$US:Xa\USnUupn [ [ XU 455n [ [ X U 455n U S:XaU S:XaUnU$[X U 4S9$[XU 4S9$Sn /nUH+n[X5(dMU S- n URU5 M- US :Xa U S:XaWnU$US- U s=::aU::axO Ou[5nUHnUHnURU5 M M UH5nUH,nUU:wdM UUU;dMS n[R"U5e M7 [U5nU$S n[R"U5e) aSelect 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. rr]r^rzstarting_edge (r`z) is not in the Graph) starting_edger,zCG 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)) rrr<rrMrer3rOrorbr/addr0)rryrcrW e_trianglesrrXrmr=r>cac_edgesbc_edgess odd_trianglestriangle_nodesr;r:rUs rrOrOs<0 aggi (  Q4qwwy ""WQqTF-#@A A Q4q1w #AaD6AaD61FGC""3' 'Q"K KAAv f e a Naz!V,-z!V,- q=1} ! P M-Q!fEE(a&A A  AQ""Q$$Q' 6a1fM2 1Ua_1_ UN"A"&&q)#$'AAv1AaD==!..s33 ($".1M  6 ""3' 'rrH)FN)__doc__ collectionsr functoolsr itertoolsrnetworkxrnetworkx.utilsrnetworkx.utils.decoratorsr__all__ _dispatchablerrrr rerorPrOr'rrrs+#",9 - .%i &i X <= @Z \"%Z &#!Z z %4P* ZXr