I would follow the next steps:

1. Check if a contains b or vice versa; if so, write the right string;

edit: check that this right string may be repeated string, like a = "dadada", b = "da"

1. Check if a have a suffix that matches some prefix of b or vice versa; i.e. a = "cd", b = "da", then write "cda"

I think that's all. All of this you may do with only prefix function.

i think printing all the characters that appears in any one of the string would work

Break the problem into 2 subproblems:

1. Find a string that contains both A and B, call it T
2. Find the smallest string S which when repeated infinitely, has T as a substring

We will solve them separately, both utilize the knowledge of Longest proper prefix/Failure function.

Task 1

1. If A is a substring of B then T = B
2. If B is a substring of A then T = A
3. T must be either A+B or B+A, but there will be some extra redundant characters. for A+B we must remove the largest prefix of B that is a suffix of A and then join A+B. Similarly for B+A. T1 = A+B, after reduction and T2 = B+A, after reduction.

Similar Idea — SPOJ — CF25E

Now we will try to form the string S from the above strings T (for T1 and T2, try both and pick the one which gives smaller S)

Task 2

We can consider the string as a circular string, it makes sense to remove the longest proper prefix that is also a suffix from T, then the leftover string when repeated infinite times will have T as the substring. Here we get the required string S.

abcab — LPS = ab, removing it gives "abc" which when repeated does give abc"abcab"c...

babab — LPS = bab, removing it gives "ab" which when repeated gives a"babab"aba...

Same Idea — A topcoder problem "Running Letters"