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It is called the empty word and is denoted by ε. 2 Consider the words u = banana, v = aaccaaa, and the empty word ε. Then, α(u) = {a, b, n} α(v) = {a, c} α(ε) = ∅ 25 26 Algorithmic Combinatorics on Partial Words The set of all words over A is denoted by A∗ and is equipped with the associative operation defined by the concatenation of two sequences. We use multiplicative notation for concatenation. For example, if u = aaa and v = bbb are words over an alphabet A, they are members of A∗ , and the word uv = aaabbb is also a member of A∗ .

In order to do this, we introduce a new symbol, , and make the following definition. 3 If u is a partial word of length n over A, then the companion of u, denoted by u , is the total function u : {0, . . , n − 1} → A ∪ { } defined by u (i) = u(i) if i ∈ D(u) otherwise We extend our definition of α(u) for any partial word u over an alphabet A in the following way: α(u) = {a ∈ A | u(i) = a for some i ∈ D(u)} It is important to remember that the symbol is not a letter of the alphabet A. Rather, it is viewed as a “do not know” symbol, and its inclusion allows us to now define a partial word in terms of the total function u given in the definition.

Prove that conjugacy Preliminaries on Partial Words 41 on full words is an equivalence relation. Is it an equivalence relation on partial words? 14 Let x be a partial word and set u = x. 14, what can be said when u = ax with x a nonempty partial word and a ∈ A? 16 S Construct a partial word u over the alphabet {0, 1} for which no a ∈ {0, 1} exists that satisfies ua is primitive. 17 H Let u ∈ W (A). For 0 < p ≤ |u|, prove that the following are equivalent: 1. The partial word u is weakly p-periodic.

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A 1-(S,T)-edge-connectivity augmentation algorithm by Enni S.

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