Is it possible to shorten this Finite Automata Transition Graph?
Each string that is input to an automaton takes you to some state. So you can see each state an encoding some kind of information about the inputs the machine has seen if it's in that state. For example, an automaton might have a state that corresponds to "all words with an even number of $b$s."

Why am I telling you this? Because, when you're designing an automaton, it's often helpful to think about what information you need to store about the input to accept the language you want. In this case, you want to accept all words whose last letter has already appeared in the string. So, the information you need to decide the language is "What letters have I seen so far?"  If you encode that information in some states of the automaton, it should be easy to add the right transitions to accept the language you want.