U 9v f?D@sdZddlmZddlmZddlmZddlZddl m Z ddl m Z dd gZ dd dZdd d ZdddZe de ddd ZddZddZddZdddZdS)z%Functions for generating line graphs.) defaultdict)partial) combinationsN)arbitrary_element)not_implemented_for line_graphinverse_line_graphcCs(|rt||d}nt|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 )Z is_directed _lg_directed_lg_undirected)Gr Lrsz"_lg_undirected..cs|d|dfS)Nrrr)Zedge node_indexrrz _lg_undirected..cs2g|]*}tt|ddjd|ddqS)Nkey)tuplesortedget)rxr"rr sz"_lg_undirected..cs g|]}tt|fdqS)r')r)r*)rb)aedge_key_functionrrr-sN) rrrrrr enumeratesetlenrupdateZadd_edges_from) r r r rrshiftrunodesrr)r/r0r#rr s$     r ZdirectedZ multigraphc sf|dkrtdS|dkrNt|}|df}|dft|fg}|S|dkrt|dkrtd}t|t|dkrd}t|t|}t ||}dd|j D|D]}|D]}|d7<qqt dkrd}t|t fd d D} t}|||| t|j dD].\}tfd d |Dr2||q2|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)rrrr6rrrr!$sz&inverse_line_graph..r&zEG is not a line graph (vertex found in more than two partition cells)c3s |]}|dkr|fVqdS)rNrr8)P_countrr ,s z%inverse_line_graph..c3s|]}|kVqdS)Nr)rZa_bit)r.rrr:1s)Znumber_of_nodesrrrZGraphnumber_of_edges NetworkXErrorZnumber_of_selfloops_select_starting_cell_find_partitionr7maxvaluesr)Zadd_nodes_fromranyr) r vr/Hmsg starting_cellPpr6Wr)r9r.rrsB4         cCsx|\}}||kr"td|d|||krFtd|d|dg}||D] }|||krR||||fqR|S)z.Return list of all triangles containing edge eVertex not in graphEdge (, ) not in graph)rr<append)r er6rBZ triangle_listr,rrr _triangles6s   rPcs|D]"}||krtd|dqtt|dD]8}|d||dkr6td|dd|ddq6tt|D]*}||D]}||kr|d7<qq|tfd d DS) aTest whether T is an odd triangle in G Parameters ---------- G : NetworkX Graph T : 3-tuple of vertices forming triangle in G Returns ------- True is T is an odd triangle False otherwise Raises ------ NetworkXError T is not a triangle in G Notes ----- An odd triangle is one in which there exists another vertex in G which is adjacent to either exactly one or exactly all three of the vertices in the triangle. rIrJr&rrrKrLrMc3s|]}|dkVqdS))rNr)rrBZ T_neighborsrrr:isz _odd_triangle..)r7rr<listrrintrA)r Tr6rOtrBrrRr _odd_triangleDs " rWc Cs|}|g}|tt|dt|}|dkr|}t||}|dkr*|gt||}|D]0}|D]&}||krp|||krpd} t| qpqh| t ||tt|d||7}q*|S)aiFind a partition of the vertices of G into cells of complete graphs Parameters ---------- G : NetworkX Graph starting_cell : tuple of vertices in G which form a cell Returns ------- List of tuples of vertices of G Raises ------ NetworkXError If a cell is not a complete subgraph then G is not a line graph r&rz=G is not a line graph(partition cell not a complete subgraph)) copyZremove_edges_fromrSrr;popr3rr<rNr)) r rEZ G_partitionrFZpartitioned_verticesr6Zdeg_uZnew_cellrBrDrrrr>ls&   r>cCs|dkrt|}nb|}|d|kr@td|dd|d||dkrxd|dd|dd}t|t||}t|}|dkr|}nf|dkr|d}|\}} } tt||| f} tt|| | f} | dkr| dkr|}nt|| | fd Snt||| fd Snd} g}|D]$}t||r"| d7} | |q"|d krb| dkrb|}n|d| kr~|krnnpt }|D]}|D]}| |qq|D]8}|D],}||kr|||krd }t|qqt |}nd }t||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. NrrIrJrzstarting_edge (rLz) 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)) rrr7rr<rPr3r=rWrNr2addr))r rZrOrDZ e_trianglesrrErUr/r.cZac_edgesZbc_edgessZ odd_trianglesZtriangle_nodesr,r6rBrrrr=s\         r=)N)N)FN)N)__doc__ collectionsr functoolsr itertoolsrZnetworkxrZnetworkx.utilsrZnetworkx.utils.decoratorsr__all__rr r rrPrWr>r=rrrrs"      l  @ ](-