Ultimately, and I'm still thinking about this.... I think you need to do 2 things.
1. Create an adjacency list class which can store the information in a thoughtful way. You need to store destination cities and cost, right? So I think your DestinationList class might have an attribute that is a LinkedList and each index can hold a
String[2], where index 0 is the destination and index 1 is the cost. I would still have used a Map for this, but if the requirement is a LinkedList, then so be it. The method, addDestination(), can create the array from a String and an Integer and append it to the end of the LinkedList. You can also have methods removeDestination(), findDestination(), showDestinations().... ad nauseum depending on how far you want to take this thing.
2. Store that list in a TreeMap, HashMap, or some other kind of Map that stores the name of the origin city as the KEY, and the DestinationList as the VALUE.... look at the API for TreeMap or HashMap for how to do that.
I still don't see a reason to store anything in an object... other than the DestinationList..... for this application.