I know that a graph G' is a subgraph for G if V(G')V(G) and E(G')E(G).Today my professor wrote that G' is a subgraph for G if G' is a graph and V(G')V(G), and then he told us that this definition is equivalent to the one we've seen before (= the first line in this post).However, I don't think that these definitions are equivalent. In fact, if this is G:       2     /   \    1     3and this is G':    1 - 3we have that G' is a subgraph for G according to the second definition (wrong), while it is not according to the first one (right).Am I missing something or was the professor wrong?
Equivalent definitions for subgraph
graphs
The alternative definition that you attribute to your professor is wrong, as your counterexample clearly shows.I hope your professor actually said something else and you misunderstood him/her!