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| """ | |
| Ramsey numbers. | |
| """ | |
| import networkx as nx | |
| from networkx.utils import not_implemented_for | |
| from ...utils import arbitrary_element | |
| __all__ = ["ramsey_R2"] | |
| def ramsey_R2(G): | |
| r"""Compute the largest clique and largest independent set in `G`. | |
| This can be used to estimate bounds for the 2-color | |
| Ramsey number `R(2;s,t)` for `G`. | |
| This is a recursive implementation which could run into trouble | |
| for large recursions. Note that self-loop edges are ignored. | |
| Parameters | |
| ---------- | |
| G : NetworkX graph | |
| Undirected graph | |
| Returns | |
| ------- | |
| max_pair : (set, set) tuple | |
| Maximum clique, Maximum independent set. | |
| Raises | |
| ------ | |
| NetworkXNotImplemented | |
| If the graph is directed or is a multigraph. | |
| """ | |
| if not G: | |
| return set(), set() | |
| node = arbitrary_element(G) | |
| nbrs = (nbr for nbr in nx.all_neighbors(G, node) if nbr != node) | |
| nnbrs = nx.non_neighbors(G, node) | |
| c_1, i_1 = ramsey_R2(G.subgraph(nbrs).copy()) | |
| c_2, i_2 = ramsey_R2(G.subgraph(nnbrs).copy()) | |
| c_1.add(node) | |
| i_2.add(node) | |
| # Choose the larger of the two cliques and the larger of the two | |
| # independent sets, according to cardinality. | |
| return max(c_1, c_2, key=len), max(i_1, i_2, key=len) | |