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...mic the harder it is to extend it to a minimal-palindromic word (a strict definition is given in the next section). In the end of their paper, Holub and Saari posed a few plausible-looking questions, which would have, if answered positively, made computations of MP-ratios significantly simpler. The aim of this paper is to answer these questions, and also one further question of a similar kind. Rather surprisingly, all the answers turn out to be negative. 2 Preliminaries Let us present the notation and necessary definitions. In case of any unclear notions, we direct the reader to, for example, [4]. Given a set Σ called the alphabet, we call its elements letters, and finite sequences of letters are called words. For words w = a1a2 . . . an and u = b1b2 . . . bm (where a1, . . . , an, b1, . . . , bm ∈ Σ), with wu we denote the concatenation of words w and u, that is, wu = a1a2 . . . anb1b2 . . . bm. Given a word w and an integer k > 0, we write wk for ww . . . w︸ ︷︷ ︸ k times (called the k-th power of a word w). 2 The set of all words over the alphabet Σ is denoted with Σ∗. If a is a letter, we write a∗ for the set {ak : k > 0}, and if b is an additional letter, we write a∗b∗ for the set...

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... word extension. 1 Introduction Combinatorics on words is a branch of mathematics having a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Clearly, different approaches can be imagined, depending on the interpretation of “more palindromic”. Holub and Saari [9] chose the following one. Restricting themselves to the binary words, they observe...

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...e 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 Keywords: (scattered) subword, palindrome, word extension. 1 Introduction Combinatorics on words is a branch of mathematics having a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defini...

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...on Combinatorics on words is a branch of mathematics having a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Clearly, different approaches can be imagined, depending on the interpretation of “more palindromic”. Holub and Saari [9] chose the following one. Restricting themselves to the binary words, they observed that each word w contains ...

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...e 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 Keywords: (scattered) subword, palindrome, word extension. 1 Introduction Combinatorics on words is a branch of mathematics having a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defini...

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... word extension. 1 Introduction Combinatorics on words is a branch of mathematics having a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Clearly, different approaches can be imagined, depending on the interpretation of “more palindromic”. Holub and Saari [9] chose the following one. Restricting themselves to the binary words, they observe...

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... word extension. 1 Introduction Combinatorics on words is a branch of mathematics having a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Clearly, different approaches can be imagined, depending on the interpretation of “more palindromic”. Holub and Saari [9] chose the following one. Restricting themselves to the binary words, they observe...

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...on Combinatorics on words is a branch of mathematics having a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Clearly, different approaches can be imagined, depending on the interpretation of “more palindromic”. Holub and Saari [9] chose the following one. Restricting themselves to the binary words, they observed that each word w contains ...

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...ing a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Clearly, different approaches can be imagined, depending on the interpretation of “more palindromic”. Holub and Saari [9] chose the following one. Restricting themselves to the binary words, they observed that each word w contains a palindromic (scattered) subword of length at least ⌈ |w|...

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...ing a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Clearly, different approaches can be imagined, depending on the interpretation of “more palindromic”. Holub and Saari [9] chose the following one. Restricting themselves to the binary words, they observed that each word w contains a palindromic (scattered) subword of length at least ⌈ |w|...

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...her have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Clearly, different approaches can be imagined, depending on the interpretation of “more palindromic”. Holub and Saari [9] chose the following one. Restricting themselves to the binary words, they observed that each word w contains a palindromic (scattered) subword of length at least ⌈ |w| 2 ⌉ : a subword consisting of the dominant letter. On the basis of this observation, they called words w which do not contain palindromic subwords of length greater than ⌈ |w| 2 ⌉ minimal-palindromic: intuitively, these are the least palindromic words. The degree of “palindromicity” of a word w is then measured by the so-called MP-ratio, defined with the following conception in mind: the word is more palindromic the harder it i...

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...ing a very wide scope of applications. A similar thing can be said about palindromic words, that is, words that can be read indistinctly from left to right or from right to left. Namely, they play a major role in the study of so-called Sturmian sequences [12, 7], which in turn have applications in number theory, routing 1 optimization, computer graphics and image processing, pattern recognition and more [1, Chapter 9]. Palindromes further have applications in seemingly unrelated fields such as quantum physics [8, 2, 6], molecular biology [11, 10] [13, Chapter 4] and recently even music theory [14, 5, 3]. Thus, a more detailed knowledge about the behavior of palindromes is of a growing importance. One of the questions arising is determining which of two given words (not necessarily palindromes) is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Clearly, different approaches can be imagined, depending on the interpretation of “more palindromic”. Holub and Saari [9] chose the following one. Restricting themselves to the binary words, they observed that each word w contains a palindromic (scattered) subword of length at least ⌈ |w|...