Great question! Let's unpack this graph structure and the tools to find it:

Formal Names for This Subgraph Type #

The structure you're describing falls into a few well-defined graph theory categories, depending on specifics:

• 2-attachment induced subgraphs: This is the most direct term. These are vertex-induced subgraphs that connect to the rest of the parent graph exclusively through two "attachment vertices" (your A and F example). The internal nodes (B-E in your case) only connect to each other or the two attachments.
• Subgraphs separated by a 2-vertex cut: A 2-vertex cut is a pair of vertices whose removal splits the graph into at least two disconnected components. The isolated component(s) that only connected to the main graph via these two vertices are exactly what you're targeting.
• For a more specific subset where the internal nodes form a biconnected component (no internal cut points), this can also be called a biconnected component with two attachment points.

Algorithms to Find These Subgraphs #

Here's a practical, step-by-step approach to locate these subgraphs, including the constraint that the shortest path between the two attachment vertices is ≤ d (e.g., 4):

1. Precompute All-Pairs Shortest Paths

   • For unweighted graphs, run BFS starting from every vertex to calculate the shortest distance between all pairs of nodes. For weighted graphs, use Dijkstra's algorithm (with a priority queue) per node, or Floyd-Warshall for smaller graphs. Store these distances in a matrix for fast lookup.

2. Detect 2-Vertex Cuts and Associated Subgraphs

   • Use a modified Tarjan's algorithm: Tarjan's original algorithm finds articulation points (single vertices whose removal disconnects the graph), but you can adapt it to detect pairs of vertices that act as a cut (2-vertex cuts).
   • For smaller graphs, a brute-force enumeration works too: iterate over all possible vertex pairs (u, v), remove them from the graph, and check if the remaining graph has disconnected components. For each component S:
     • Validate that every node in S has neighbors only within S, or only u and v (this ensures S doesn't have other connections to the main graph).

3. Filter by Shortest Path Constraint

   • For each valid attachment pair (u, v), check if their precomputed shortest path distance is ≤ d. If it meets the threshold, the associated subgraph S (plus u/v if you want to include the attachment vertices in your subgraph) is a match.

4. Avoid Duplicates

   • Ensure you don't count the same subgraph multiple times (e.g., if a component could be associated with more than one 2-vertex cut, though this is uncommon if you strictly enforce the "only two attachments" rule).

内容的提问来源于stack exchange，提问作者LizzAlice