/sci/ - Science & Math

I'm confused on the definition of subgraph.
We define a graph as G=(V,E) with V being a finite set and E satisfying [math]E\subseteq [V]^{2}[/math]. We can allow E to be the empty set, but V will always be not empty.
G'=(V',E') is a subgraph of G if [math]V'\subseteq V[/math] and [math]E'\subseteq E[/math].
So, if in this picture we use the red points to form V' and E' only has the diagonal, that would be a valid subgraph, even though by itself it would not be a graph since [math]E'\nsubseteq [V']^{2}[/math]. Am I wrong somewhere? Because if I'm not, I don't see the point of such a subgraph.