Find all the ways to choose $k$ objects from a list of $n$ objects (using a graph?)
To generalize your approach to $k$-subsets of an $n$-set, you would need to build a hypergraph. Ordinary graph edges are relations between pairs of vertices. Hypergraphs allow relations between arbitrary sets of vertices. They are extremely general objects, essentially representing families of sets.
This question provides two good answers for how to (reasonably) efficiently enumerate these subsets. The first uses bit magic and is a special case when $n$ is small enough. The second takes a recursive approach which saves having to compute the first $k-1$ elements of a $k$ subset multiple times, but only really helps if enumerating over several consecutive values of $k$.