Counter-intuitive answers to some questions concerning minimal-palindromic extensions of binary words

In [Š. Holub & K. Saari, On highly palindromic words, Discrete Appl. Math. 157 (2009), 953–959] the authors proposed to measure the degree of “palindromicity” of a binary word w by ratio |rws| |w| , where the word rws is minimal-palindromic—that is, does not contain palindromic subwords of length greater than ⌈ |w| 2 ⌉ —and the length |r|+|s| is as small as possible. It was asked whether the words of a given length n which reach the maximal possible ratio |rws| |w| among the words of length n are always palindromes. It was further asked whether it can be assumed, w.l.o.g., that r and s are of form 0∗ or 1∗, or at least 0∗1∗ or 1∗0∗. We negatively answer these questions, and also one further question of a similar kind. Mathematics Subject Classification (2010): 68R15