How to handle test case dependency when one test must use the functionality tested by another

I am trying out unit testing in Java with JUnit for an assignment. Previously, we were tasked with creating a generic puzzle-solving algorithm using graph searching algorithms.

The important part is that I have an interface called Graph Vertex.

public abstract class GraphVertex { 
abstract public List<GraphVertex> getAdjacentVertices();
    abstract boolean equals(Object o);
    abstract int hashCode();
}

This allows the user to swap between different node types if he so wishes.

Now, I have a concrete class called SliderPuzzleVertex<T> that extends GraphVertex, and holds a 2-dimensional array of T and a set of positions (x, y) that are considered empty. The getAdjacentVertices generates new SliderPuzzleVertex-es of every possible move that can be made.

I need to test getAdjacentVertices for this class.

Using Integer as my parameterized T, I initialized an instance and called the method, and now I am left with a List<GraphVertex>, but I want to test:

1. That they are all SliderPuzzleVertex.

2. Their fields (for example, that the size if their position set remained the same),

3. Specific cases where i give the input and expected output, and check that they are same using equals().

You can easily see that in order to test 2 I would have to cast them to SliderPuzzleVertex, therefore relying on test 1. This test case dependency is what I am trying to avoid

How can I approach this problem?