Suppose we have a graph g and a subset of vs = VertexList[g]. The vertex names of g could be anything.

How can I quickly and reliably get the subgraph induced by vs?

Subgraph[g, vs] does not work reliably. Consider the following example:

Let's take a list of sets,

sets = {{2}, {3}, {2, 3}}

Their elements are

elements = Union @@ sets
(* {2, 3} *)

Now let us build the containment-relationship graph:

g = RelationGraph[MemberQ, Join[sets, elements]]

This is a subset of it's elements:

vs1 = {2,3}

SubsetQ[VertexList[g], vs1]
(* True *)

Subgraph does not give what I need and described above:

Subgraph[g, vs1, VertexLabels -> Automatic]

Getting the subgraph of a larger graph is a fundamental operation. Is it possible in Mathematica?

Note that when developing a package, it is not possible to control the input that the package functions receive. I need a way that is fast and that works with any graph and any subset of its vertices. A solution that works only with the example graph from above is not satisfactory.

I posted this question because I am hoping that Subgraph has some way to disambiguate its input.

Extract has a potentially ambiguous syntax. But it's not really ambiguous. Extract[list, 1] takes the first element. What does Extract[list, {1,2}] do? it takes element 2 at level 1, and not the first two elements. For that we can use Extract[list, {{1}, {2}]]. The unambiguous way is Extract[..., {pos1, pos2, ...}] where each pos is a List. This is a robust syntax.

Lookup has a potentially ambiguous syntax. But again there's a completely unabiguous version: use a list as the second argument. asc = <|{1, 2} -> "a", 1 -> "b", 2 -> "c"|>;, then Lookup[asc, {1,2}] gives the results for keys 1 and 2 and Lookup[asc, {{1,2}}] gives it for key {1,2} in a list. I can program confidently with this. There's also Key if I want a single result.

Subgraph is complicated because the second argument can be either a list of vertices or a pattern, which is an extremely general thing. Is there any way out or is the design just broken?

VertexDelete suffers from the same problem.