o h{D@sdZddlmZddlmZddlmZddlZddl m Z ddl m Z dd gZ ejd d dd dZdd dZdddZe de dejd d dd ZddZddZddZdddZdS)z%Functions for generating line graphs.) defaultdict)partial) combinationsN)arbitrary_element)not_implemented_for line_graphinverse_line_graphT) returns_graphcCs*|r t||d}|St|d|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 Lrl/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/networkx/generators/line.pyrs f cCsftjd||jd}|rt|jddn|j}|D]}||||dD]}|||q'q|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_noderrrr {s  r Fc stjd||jd}|rt|jddn|j}|rdnd}ddt|Dfdd t}|D]6}fd d ||D}t|dkrK| |dt|D]\} | fd d || |d DqOq1| ||S)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. rrTrrcSsi|]\}}||qSrr).0inrrr sz"_lg_undirected..cs|d|dfS)Nrrr)edge node_indexrrsz _lg_undirected..cs2g|]}tt|ddjd|ddqS)Nkey)tuplesortedget)r"xr'rr s2z"_lg_undirected..cs g|] }tt|fdqS)r+)r-r.)r"b)aedge_key_functionrrr1sN) rrrrrr enumeratesetlenrupdateadd_edges_from) rr r rrshiftrunodesr#r)r3r4r(rrs&     rdirected multigraphc sb|dkr tdS|dkr't|}|df}|dft|fg}|S|dkr:|dkr:d}t|t|dkrHd}t|t|}t ||}dd|j D|D]}|D] }|d7<q_q[t dkrzd}t|t fd d D} t}|||| t|j dD]\}tfd d |Dr||q|S) afReturns 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.cSsi|]}|dqS)rrr"r;rrrr%'sz&inverse_line_graph..r*zEG is not a line graph (vertex found in more than two partition cells)c3s"|] }|dkr|fVqdS)rNrr?)P_countrr /s z%inverse_line_graph..c3s|]}|vVqdSNr)r"a_bit)r2rrrA4s)number_of_nodesrrrGraphnumber_of_edges NetworkXErrornumber_of_selfloops_select_starting_cell_find_partitionr<maxvaluesr-add_nodes_fromranyr) rvr3Hmsg starting_cellPpr;Wr)r@r2rrsF 5         cCsx|\}}||vrtd|d|||vr#td|d|dg}||D]}|||vr9||||fq)|S)z.Return list of all triangles containing edge eVertex not in graphEdge (, ) not in graph)rrGappend)rer;rO triangle_listr0rrr _triangles9s   r^cs|D]}||vrtd|dqtt|dD]}|d||dvr7td|dd|ddqtt|D]}||D]}||vrR|d7<qDq>tfd d 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. rVrWr*rrrXrYrZc3s|] }|dvVqdS))rNr)r"rOT_nbrsrrrAlsz _odd_triangle..)r<rrGlistrrintrN)rTr;r\trOrr`r _odd_triangleGs    rfc Cs|}|g}|tt|dt|}|dkrh|}t||}|dkrb|gt||}|D]}|D]}||krK|||vrKd} t| q8q4| t ||tt|d||7}|dks|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 r*rz>G is not a line graph (partition cell not a complete subgraph)) copyremove_edges_fromrbrrFpopr7rrGr[r-) rrR G_partitionrSpartitioned_verticesr;deg_unew_cellrOrQrrrrJos,    rJcCs|dur t|}n1|}|d|vr td|dd|d||dvrs(        l @ ](-