[spqr_tree] Per-node planarity testing + virtual-edge skeleton embeddings

[devin-ai-integration]

Summary

Adds planarity testing and combinatorial embedding to spqr_tree, computed per SPQR node as each skeleton is sealed (in finalize_node), not as a separate pass.

New API:

• nodes[n].planar (only R skeletons can fail), blocks[b].planar, is_planar (conjunctions).
• embed_next: a rotation system over virtual-edge half-edges, node by node — half-edge 2*ve+k is vedges[ve]'s endpoint at vs[k], embed_next[h] is the next half-edge around that skeleton vertex. Glue across nodes via vedges[ve].o_ve (R rotations unique up to reflection, P freely permutable, S/Q/I/O rigid).

How: each sealed vedge range is already a DFS of the node's skeleton (every tree vedge appears after its whole subtree; every vedge is a collapsed 2-attachment component, so it acts as a single tree/back edge). emit_node_embedding handles Q/I/O/S/P trivially; emit_rigid does four passes over the range: index skeleton vertices → recompute skeleton lowpoints bottom-up + re-sort each vertex's outgoing vedges by skeleton nesting depth (collapsing a subtree can change its lowpt/chordality, so the original bucket-sorted order is not a valid nesting order — this was verified empirically) → Brandes' left-right test on the skeleton (conflict-pair stack, implicit two-coloring via lr_ref/lr_side) → sign resolution + rotation emission. A non-planar R skeleton only clears its own flag and falls back to an arbitrary (still circular) rotation; the rest of the block is unaffected.

Design notes in doc/spqr_planarity.md.

Validation

• Existing structural tests + new per-node checks (rotation at each skeleton vertex is a circular permutation of exactly its incident half-edges; every planar node satisfies V - E + F == 2, i.e. genus 0) pass under ASan/UBSan, including 4000 extra random multigraphs up to 60 vertices / 140 edges.
• Fixtures: K4, K5, K5−e, K3,3, Petersen, cube, multigraphs with self loops/parallels/bridges; K5 + square through a cutvertex (per-block flags); K5 and K4 sharing an edge (one block, two R nodes, only one non-planar).
• is_planar matches networkx on 60,000 random multigraphs.

Link to Devin session: https://app.devin.ai/sessions/8d675c3c6ba24d019643c3fabdd14e23
Requested by: @ecnerwala

---

[devin-ai-integration]

🤖 Devin AI Engineer

I'll be helping with this pull request! Here's what you should know:

✅ I will automatically:

• Address comments on this PR. Add '(aside)' to your comment to have me ignore it.
• Look at CI failures and help fix them

Note: I can only respond to comments from users who have write access to this repository.

⚙️ Control Options:

• Disable automatic comment, CI, and merge conflict monitoring

[beta-devin-ai-integration]

✅ Devin Review: No Issues Found

Devin Review analyzed this PR and found no bugs or issues to report.