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On the Relation between Periodicity and Unbordered Factors of Finite Words

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On the Relation between Periodicity and Unbordered Factors of Finite Words
Finite words and their overlap properties are considered in this paper. Let w be a finite word of length n with period p and where the maximum length of its unbordered factors equals k. A word is called unbordered if it possesses no proper prefix that is also a suffix of that word. Suppose k < p in w. It is known that n 2k - 2, if w has an unbordered prefix u of length k. We show that, if n = 2k - 2 then u ends in abi , with two different letters a and b and i 1, and bi occurs exactly once in w. This answers a conjecture by Harju and the second author of this paper about a structural property of maximum Duval extensions. Moreover, we show here that i < k/3, which in turn leads us to the solution of a special case of a problem raised by Ehrenfeucht and Silberger in 1979.
Stepan Holub, Dirk Nowotka
Added 29 Oct 2010
Updated 29 Oct 2010
Type Conference
Year 2008
Where DLT
Authors Stepan Holub, Dirk Nowotka
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