A common definition of a graph (for instance found in Diestel‘s Graph Theory) is the following:
A graph is a pair $(V,E)$ with $E \subseteq [V]^2$.
However, this definition allows edges to be vertices and vice versa: Let $V$:= {1,2,{1,2}} and $E$={{1,2}}. Then $(V,E)$ is a graph.
Drawing a graph by representing vertices as dots and edges as lines between dots seems not to work here. This contradicts my intuition in working with graphs.
Obviously, in the above example there exists a graph $G'=(V',E')$ isomorphic to $G$ such that $V' \cap E' = \emptyset$. It seems that there is always such graph.
Is there any reason why one does not require $V \cap E = \emptyset$ to hold?
Simply because given a graph $G$ there is always an isomorphic graph $G'= (V',E')$ such that $V' \cap E' = \emptyset$? Is there anything more behind it than that?