A combinatorial view on string attractors

The notion of emph{string attractor} has recently been introduced in [Prezza, 2017] and studied in [Kempa and Prezza, 2018] to provide a unifying framework for known dictionary-based compressors. A string attractor for a word $w=w_1w_2cdots w_n$ is a subset $Gamma$ of the positions ${1,ldots,n}$, such that all distinct factors of $w$ have an occurrence crossing at least one of the elements of $Gamma$. In this paper we explore the notion of string attractor by focusing on its combinatorial properties. In particular, we show how the size of the smallest string attractor of a word varies when combinatorial operations are applied and we deduce that such a measure is not monotone. Moreover, we introduce a circular variant of the notion of string attractor to provide a characterization of the conjugacy classes of standard Sturmian words.