Can Fibonacci Heap have more than one nodes with equal rank(or value, or key)?

Question

I am studying Fibonacci heap alone, and I came across a question.

I know any nodes can be inserted into Fibonacci heap, but what if the rank(or value, or key) of that new node is equal to the sibling node?

1) For example,

   (1) <-minimum root
  /   \
(3)   (5)

and what happens if node of (1) is inserted?

(1) --- (1)
       /   \
     (3)   (5)

2) Or what happens in this kind of situation?

before:

 (2)-----(4)-----(5)
  |      / \     / \
 (1)   (6) (7) (8) (9)

after:

 (2)-----(4)-----(5)    +     (5)
  |              / \          / \
 (1)           (8) (9)      (1) (2)

Where would the new node-tree belong?

3) How would you consolidate(or order) this kind of tree - that has a pair of equal nodes at the root?

 (10) ---- (10) ----- (1) ----- (3) ----- (4)

I'd be grateful if anyone can help me solve this little confusion on Fibonacci heap. Thank you.

Answer 1

Nodes with equal keys are OK.

Recall the heap property: for min-heap, the key of each node is greater than or equal to the key of its parent. (The max-heap property can similarly defined.)

A Fibonacci heap is just a collection of such trees. If more than one root has the minimum key, then any such root can be the minimum node.

In you are examples, there's nothing wrong with (1) and (3). But in (2), you have trees that violate the min-heap property.