Unavoidable Sets of Partial Words of Uniform Length

A set X of partial words over a finite alphabet A is called unavoidable if every two-sided infinite word over A has a factor compatible with an element of X. Unlike the case of a set of words without holes, the problem of deciding whether or not a given finite set of n partial words over a k-letter alphabet is avoidable is NP-hard, even when we restrict to a set of partial words of uniform length. So classifying such sets, with parameters k and n, as avoidable or unavoidable becomes an interesting problem. In this paper, we work towards this classification problem by investigating the maximum number of holes we can fill in unavoidable sets of partial words of uniform length over an alphabet of any fixed size, while maintaining the unavoidability property.

💡 Research Summary

The paper investigates the combinatorial structure of unavoidable sets of partial words when all words have the same length m. A partial word is a finite string over a finite alphabet A in which some positions are “holes” (denoted by ⋄) that can match any letter. An infinite two‑sided word w meets a partial word u if some factor of w is compatible with u; otherwise w avoids u. A set X of partial words is called unavoidable if every infinite word over A meets at least one element of X.

The authors first recall that, unlike the hole‑free case where avoidance can be decided in polynomial time, determining unavoidability for partial words is NP‑hard even when all words have the same length. Consequently, classifying unavoidable sets with respect to the alphabet size k and the number of words n becomes a natural challenge.

To formalize the quantitative aspect, they introduce the function H_{k,m,n}: the minimum total number of holes among all unavoidable m‑uniform sets of size n over a k‑letter alphabet. Prior work