**Abstract**

Partial words are sequences over a finite alphabet that may have holes that match, or are compatible with, all letters in the alphabet;
partial words without holes are simply words. Given a partial word
*w*, we denote by *sub _{w}(n)* the set of subwords of

*w*of length

*n*, that is, words over the alphabet that are compatible with factors of

*w*of length

*n*. It was shown that for each non-negative integer

*h*and each positive integer

*N*, there exists a binary partial word

*w*with

*h*holes such that

*sub*has size

_{w}(n)*2n*for all

*n = 1, 2, ..., N*. Here, we consider the classification of all such partial words of minimal length. Our motivation is the construction of binary de Bruijn partial words of order

*n*, which are partial words over the binary alphabet

*{a,b}*of minimal length containing all the length

*n*subwords; these partial words are said to represent the set

*{a,b}*. More generally, a partial word

^{n}*w*represents a set

*S*of words of equal length

*n*if

*S = sub*. Using a graph theoretical approach, we also investigate some computational problems related to this concept of representability.

_{w}(n)**Keywords**

Combinatorics on Words; Graph Theory; Algorithms; Partial Words; Subwords; Representable Sets.