How do I change the Head of Graph into List?

Question

I am trying to change the Head of b (a NeighborhoodGraph) from Graph into List but Apply seems to fail. I start with a list of connected vertices:

a={1\[UndirectedEdge]1,2\[UndirectedEdge]2,1\[UndirectedEdge]3,3\[UndirectedEdge]3,
4\[UndirectedEdge]4,3\[UndirectedEdge]4,5\[UndirectedEdge]5,6\[UndirectedEdge]6,
6\[UndirectedEdge]2,7\[UndirectedEdge]7,8\[UndirectedEdge]5,8\[UndirectedEdge]8,
9\[UndirectedEdge]9,10\[UndirectedEdge]10,10\[UndirectedEdge]4};

I calculate the neighborhood graph of node 3. Its Head is Graph:

b=NeighborhoodGraph[a//Graph,3]
b//Head
(* ==> Graph *)

I use Apply to change the Head to List:

c=Apply[List,b];

However, the Head of c is still Graph:

c//Head
(* ==> Graph *)

My real question is how to access the list of edges in a NeighborhoodGraph.

Answer 1

Apply does not work on Graph objects because they are atomic (you can verify this using AtomQ).

You can use EdgeList to retrieve the edge list of any graph.

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To get access to the compound representation of an atomic Graph, you can use the general techniques described here and here. I recommend using Carl Woll's Nucleus, which wraps this up into a nice easy-to-use function. Be warned that manipulating this representation of Graph takes you into undocumented territory. Therefore, try to avoid it if you can. Sometimes it's necessary though.

Answer 2

Update due to valid comment by Szabolcs. I create the Graph

$$BuckyBall = ResourceFunction["BuckyballGraph"][0, VertexCoordinates -> "Embedded"]

I disassemble $$BuckyBall:

    {$$BBVertexList, $$BBVertexCount, $$BBEdgeList, $$BBVertexCoordinates} =
         {$$BuckyBall // VertexList, $$BuckyBall // VertexCount,
$$BuckyBall // EdgeList, $$BuckyBall // GraphEmbedding}

Work on the vertex coordinates and then re-assemble as Graph3D:

Graph3D[$$BBVertexList,$$BBEdgeList,VertexCoordinates->$$BBVertexCoordinates]

---

Original post. Still valid, but undoubtedly a desperate hack.

$$BuckyBall is the name of a Graph

 $$BB = StringReplace[$$BuckyBall // InputForm // ToString, 
    "Graph[" -> "List[", 1] // ToExpression;

Any simplification is appreciated. The term , 1] is important, otherwise all "Graph" instances are replaced! Valid Graph expressions are created with

GraphicsRow[{Graph[$$BB[[1]], $$BB[[2]], $$BB[[3]]], 
  Graph[Sequence @@ $$BB], Graph @@ $$BB}]

I use this to extract the list of vertex coordinates, transform/manipulate them and replace the list of vertex coordinates in a copy of $$BB.