implemented function for acyclic orientations

[saatvikraoIITGN]

Description
This PR introduces an efficient algorithm for generating all acyclic orientations of an undirected graph based on the work by Mathew B. Squire. The algorithm utilizes a recursive approach along with concepts from posets and subsets to improve the efficiency of acyclic orientation generation significantly.

Changes Made

1. Implemented the reorder_vertices function to efficiently reorder vertices based on a specific criterion, improving subsequent acyclic orientation generation.
2. Created the order_edges function to assign labels to edges based on the reordered vertices, simplifying the acyclic orientation process.
3. Introduced the generate_orientations function to efficiently generate acyclic orientations by considering subsets of edges and checking for upsets in a corresponding poset.
4. Implemented the recursive helper function to generate acyclic orientations for smaller graphs, building up to larger graphs.
5. Modified the main function to use SageMath for graph manipulation and incorporated the new algorithm.

Fixes #37253

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[grhkm21]

Is this related to #35075? Also, lint CI is failing.

[dcoudert]

Thank you for this first draft. I see that there is room for improvements. A first issue is that you implicitly assume that the vertices of the input graph are labeled 0..n-1 which is not always the case. To cope with this issue, you should use method relabel to get both a copy of the input graph with suitable vertex labels and the mapping.

Below are a few comments.

[dcoudert]

To fix the failing doctests, you must expose the method in Graph. See the bottom of file src/sage/graphs/graph.py to understand how to do that.

[dcoudert]

We still have serious issues. We also have some minor issues regarding the coding style (too many parenthesis in unnecessary places, iteration of G.vertices() when over G is enough, etc.), but this can be fixed in a separate PR.

[dcoudert]

Can you add doctests for small cases and check that the output is consistent. I don't think it is so far.

sage: list(Graph().acyclic_orientations())
[Graph on 0 vertices]
sage: list(Graph(1).acyclic_orientations())
[Graph on 0 vertices]
sage: list(Graph(2).acyclic_orientations())
[Graph on 0 vertices]

[dcoudert]

A graph without edge has no orientation, right ? so the method should not return an empty graph is such case.

[dcoudert]

You could add at the beginning of the method:

if not G.size():
    # A graph without edge cannot be oriented
    return

[dcoudert]

Don't forget that the output of the doctests are now DiGraph(...) and not Graph(...).

[dcoudert]

LGTM.

We can certainly improve the code to get a faster method, but in a future PR. I will think about it.

[github-actions]

Documentation preview for this PR (built with commit bef9946; changes) is ready! 🎉