Given a graph and a subset of its vertices, how to get an induced subgraph by customizing?

Question

An induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges (from the original graph) connecting pairs of vertices in that subset. For example:

Graph[{"A" <-> "B", "A" <-> "C", "A" <-> "F", "B" <-> "D", 
  "B" <-> "E", "D" <-> "E", "A" <-> "E", "B" <-> "F"}, 
 VertexLabels -> Automatic]

There is only one subgraph induced by {A, B, D, E} in the above graph:

The InduceSubgraph function does not seem to be directly available from Mathematica.

• https://reference.wolfram.com/language/Combinatorica/ref/InduceSubgraph.html

When I use the Combinatorica package, it breaks mathematica's own Graph function.

So I wanted to get the induced subgraph by extracting sub-matrix. But I am not very clear how the vertex labels of the graph correspond to the labels of its adjacency matrix. So the code below is elementary and not very reliable.

Inducedgraph[g_, vlist_] := Module[{sub}, AdjacencyMatrix[g];
   sub = s[[vlist, vlist]];
   AdjacencyGraph[sub]];

Especially when vertices of a graph are labeled with letters rather than numbers, it seems more important to find reliable code.

Edit: Thanks yode for the reminder that Subgraph can do that. But I'm also interested in a custom implementation of this function.

Answer 1

g1 = Graph[{"A" <-> "B", "A" <-> "C", "A" <-> "F", "B" <-> "D", "B" <-> "E", "D" <-> "E", "A" <-> "E", "B" <-> "F"}, VertexLabels -> Automatic]  
vlist={"A", "B", "D", "E"}

Using the built in function:

Subgraph[g1, vlist]

Custom Function: (Creates a graph from edges where both vertices are in the list)

InducedGraph[g_, v_] := Module[{},
  Graph[Select[EdgeList[g], 
    MemberQ[v, First[#]] && MemberQ[v, Last[#]] &], 
   VertexLabels -> Automatic]]
InducedGraph[g1, vlist]