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

Abstract

In [. Holub, K. Saari, On highly palindromic words, Discrete Appl. Math. 157 (2009) 953959] the authors proposed to measure the degree of "palindromicity" of a binary word w by ratio |rws

w|, where the word rws is minimal-palindromicthat 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. © 2011 Elsevier B.V. All rights reserved.