Rearranging strings so that the Hamming distance between them is 1

Question

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Given an array of equal-length strings, check if it is possible to rearrange the strings in such a way that after the rearrangement the strings at consecutive positions would differ by exactly one character.

Example

• For inputArray = ["aba", "bbb", "bab"], the output should be stringsRearrangement(inputArray) = false; All rearrangements don't satisfy the description condition. (sic)

• For inputArray = ["ab", "bb", "aa"], the output should be stringsRearrangement(inputArray) = true. Strings can be rearranged in the following way: "aa", "ab", "bb".

I found an exponential-time algorithm, but is there a polynomial-time algorithm? Is this problem NP-complete?

I know that a closely related problem, the Hamming Salesman Problem (HTSP), is NP-complete, and this problem is certainly reducible to HTSP, but I don't know if HTSP is reducible to this one.