Computing theory: can a single node be a subgraph?
What about the following argument?
  
  Assume a single node can be considered a graph. Any graph is an induced subgraph of itself. Therefore, a single node graph has a single-node induced subgraph.
  
  Though this is only valid if a single node can be considered a graph.


That's completely circular. If a single node can be a graph, you can ask about its subgraphs and, sure, every graph is a subgraph of itself. But if a single node can't be a graph, it doesn't make sense to ask about subgraphs of something that isn't a graph.

In general, a single vertex is considered to be a graph, referred to as the "trivial graph".  However, it's something of a special case in that it's often an exception to statements one might wish to make when proving things.  For example, every connected graph contains at least one edge... except for the trivial graph; every graph has a proper subgraph... except for the trivial graph; etc. Because of this, writers often exclude the trivial graph from consideration. So, for example, in the "notation" section of a graph theory paper, you often see a statement such as "Except where stated otherwise, we assume that every graph contains at least one edge".

In this respect, asking whether the trivial graph is a graph is a bit like asking whether zero is a natural number. Some people will jump up and down and insist that it is; some people will jump up and down and insist that it isn't; the best plan is to say that it is or isn't according to what makes your life easiest in any particular situation.