Geodesic
On universal partial words
Discrete mathematics & theoretical computer science, Tome 19 (2017-2018) no. 1.

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A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from $A$ may contain an arbitrary number of occurrences of a special `joker' symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet $A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the subwords $000$ and $010$). We present results on the existence and non-existence of linear and cyclic universal partial words in different situations (depending on the number of $\Diamond$s and their positions), including various explicit constructions. We also provide numerous examples of universal partial words that we found with the help of a computer. 
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     title = {On universal partial words},
     journal = {Discrete mathematics & theoretical computer science},
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     year = {2017-2018},
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Chen, Herman Z. Q.; Kitaev, Sergey; Mütze, Torsten; Sun, Brian Y. On universal partial words. Discrete mathematics & theoretical computer science, Tome 19 (2017-2018) no. 1. doi : 10.23638/DMTCS-19-1-16. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-19-1-16/

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