构造性平面性测试：平面图中构建n-环的高效算法问询

Great question! That brute-force random edge-adding approach is definitely going to waste cycles with all those repeated O(n) planarity checks. Since you know the target is achievable, we can lean into the inherent structure of planar graphs to build your n-cycle without constant validation. Here are practical, efficient approaches:

1. Build Using the Planar Embedding (One-Time Calculation) #

First, compute the planar embedding of your initial graph once—this is an O(n) operation, same as the planarity test, but you only do it once. The embedding gives you the cyclic order of nodes around each face, which is the key to safe edge additions:

• For any two nodes that lie on the boundary of the same face, adding an edge between them will never break planarity. You can simply place the new edge inside that face, splitting it into two smaller faces.
• To construct your n-cycle, start with any node and traverse face boundaries to connect nodes in a way that builds up the full cycle. Since you know the target is possible, there’s a path through the face structure that lets you link all nodes into a cycle without crossing edges. Every edge you add here is guaranteed planar by the embedding, so no need for further tests.

2. Leverage Triangulation + Ear Decomposition #

If your initial graph isn’t a triangulation (every face is a triangle), first turn it into one—again, using the planar embedding, you can fill each non-triangular face with edges inside it (no planarity checks needed, since these edges stay within the face). Once you have a triangulated planar graph:

• Perform an ear decomposition (linear time) on the triangulation. This breaks the graph into a sequence of "ears" (triangles attached to the rest of the graph by two edges).
• You can then construct the n-cycle by iteratively removing ear edges and re-connecting nodes to extend the cycle. Every step here preserves planarity by design, so no validation is required. Triangulated planar graphs have well-documented methods for Hamiltonian cycle construction using this approach.

3. Target a Dominating Face #

A dominating face is a face that contains all nodes of the graph. If your initial graph already has one, you can directly use its boundary as the basis for your n-cycle—just fill in any missing edges between consecutive nodes on the face (these are safe, as they lie within the face). If not:

• Use the planar embedding to expand a face by adding edges inside adjacent faces, gradually incorporating all nodes into a single face. Each edge addition here is confined to a face, so planarity is guaranteed. Once you have a dominating face, building the n-cycle is trivial and test-free.

The core idea across all these methods is to use the planar structure once to guide all subsequent edge additions, eliminating the need for repeated planarity tests entirely.

内容的提问来源于stack exchange，提问作者Pitaya