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### Two Element Unavoidable Sets of Partial Words [chapter]

F. Blanchet-Sadri, N. C. Brownstein, Justin Palumbo
Lecture Notes in Computer Science
We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two.  ...  The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words.  ...  There, we present our definition of unavoidable sets of partial words and we introduce the problem of classifying such sets of small cardinality and in particular those with two elements, x 1 , x 2 , with  ...

### Minimum Number of Holes in Unavoidable Sets of Partial Words of Size Three [chapter]

Francine Blanchet-Sadri, Bob Chen, Aleksandar Chakarov
2011 Lecture Notes in Computer Science
This is a step towards the difficult problem of fully characterizing all unavoidable sets of partial words of size three.  ...  Partial words are sequences over a finite alphabet that may contain some undefined positions called holes. In this paper, we consider unavoidable sets of partial words of equal length.  ...  A set of partial words X over A is unavoidable if any two-sided infinite full word over A has a factor compatible with an element in X.  ...

### Testing avoidability on sets of partial words is hard

F. Blanchet-Sadri, Raphaël M. Jungers, Justin Palumbo
2009 Theoretical Computer Science
We prove that the problem of deciding whether a finite set of partial words is unavoidable is NP-hard for any alphabet of size larger than or equal to two, which is in contrast with the well-known feasability  ...  results for unavoidability of a set of full words.  ...  . • A two-sided infinite word w avoids X if no factor of w is an element of X . • The set X is unavoidable if no two-sided infinite word avoids X , that is, X is unavoidable if every two-sided infinite  ...

### Number of holes in unavoidable sets of partial words II

F. Blanchet-Sadri, Steven Ji, Elizabeth Reiland
2012 Journal of Discrete Algorithms
Towards this, we investigate the minimum size of unavoidable sets of partial words with a fixed number of holes.  ...  We are concerned with the complexity of deciding the avoidability of sets of partial words over an arbitrary alphabet.  ...  words in a set and returns bounds on the minimum cardinality of an unavoidable set with those parameters.  ...

### Unavoidable Sets of Partial Words [chapter]

2007 Discrete Mathematics and Its Applications
We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two.  ...  The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words.  ...  If a set of partial words is unavoidable, then it must have an element compatible with a factor of each two-sided infinite unary word.  ...

### Unavoidable Sets of Partial Words

F. Blanchet-Sadri, N. C. Brownstein, Andy Kalcic, Justin Palumbo, T. Weyand
2008 Theory of Computing Systems
We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two.  ...  The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words.  ...  If a set of partial words is unavoidable, then it must have an element compatible with a factor of each two-sided infinite unary word.  ...

### On the complexity of deciding avoidability of sets of partial words

2010 Theoretical Computer Science
We give a polynomial bound on the period of an infinite avoiding word, in the case of sets of full words, in terms of two parameters: the length and the number of words in the set.  ...  Palumbo, Testing avoidability on sets of partial words is hard, Theoret. Comput. Sci. 410 (2009) 968-972].  ...  If a set of partial words is unavoidable, then it must have an element compatible with a factor of each two-sided infinite unary word.  ...

### On the Complexity of Deciding Avoidability of Sets of Partial Words [chapter]

2009 Lecture Notes in Computer Science
We give a polynomial bound on the period of an infinite avoiding word, in the case of sets of full words, in terms of two parameters: the length and the number of words in the set.  ...  Palumbo, Testing avoidability on sets of partial words is hard, Theoret. Comput. Sci. 410 (2009) 968-972].  ...  If a set of partial words is unavoidable, then it must have an element compatible with a factor of each two-sided infinite unary word.  ...

### Inventories of unavoidable languages and the word-extension conjecture

Laurent Rosaz
1998 Theoretical Computer Science
In this paper, I prove that the inventory of unavoidable languages of n words can be explicitly made for every n, that the reduced unavoidable languages of given cardinality are finite in number (an unavoidable  ...  give a counterexample to the word-extension conjecture (which said that in every unavoidable language, there is a word w and a letter a, such that the language, where w is replaced by wa, is still unavoidable  ...  The set X is unavoidable if there is a jinite subset H of S such that for every element s in S -H, there are two elements u and v in S and an element x in X satisfying s = unu.  ...

### On extendibility of unavoidable sets

Christian Choffrut, Karel Culik
1984 Discrete Applied Mathematics
A subset X of a free monoid A * is said to be unavoidable if all but finitely many words in A * contain some word of X as a subword. A.  ...  This problem remains open, we give some partial solutions and show how to efficiently test unavoidability, extendibility and other properties of X related to the problem.  ...  Partial solutions In this section we consider two different conditions under which unavoidable sets are extendible.  ...

### Number of holes in unavoidable sets of partial words I

F. Blanchet-Sadri, Bob Chen, Aleksandar Chakarov
2012 Journal of Discrete Algorithms
Partial words are sequences over a finite alphabet that may contain some undefined positions called holes. We consider unavoidable sets of partial words of equal length.  ...  This is a step towards the difficult problem of fully characterizing all unavoidable sets of partial words of size three.  ...  of the D m (i, j) unavoidable sets.  ...

### Page 814 of Mathematical Reviews Vol. , Issue 86b [page]

1986 Mathematical Reviews
Given two morphisms g,h:£* — A”, their equality language E(g,h) is defined as E(g,h) = {z:z is in £* and g(z) = h(z)}. In other words, it is the set of words on which g and h are equal.  ...  A sample S is a finite set of words over the same alphabet.  ...

### Avoiding partial Latin squares and intricacy

Amanda G. Chetwynd, Susan J. Rhodes
1997 Discrete Mathematics
We also use these results to show that the intricacy of avoiding partial latin squares is two and of avoiding more general arrays is at most three.  ...  In other words, is P avoidable? We show that all 2k × 2k partial latin squares for k ~>2 are avoidable and give some results on odd partial latin squares.  ...  For n > 1, the intricacy of avoiding n × n arrays, on symbols 1,2 .... , n, in which every cell contains at most one symbol, is 2.  ...

### Computing the partial word avoidability indices of binary patterns

F. Blanchet-Sadri, Andrew Lohr, Shane Scott
2013 Journal of Discrete Algorithms
We complete the classification of binary patterns in partial words that was started in earlier publications by proving that the partial word avoidability index of the binary pattern ABABA is two and the  ...  one of the binary pattern ABBA is three.  ...  The set of all full words (resp., non-empty full words) over Σ is denoted by Σ ⁎ (resp., Σ + ), while the set of all partial words (resp., nonempty partial words) over Σ is denoted by (resp., ).  ...

### Vincular pattern avoidance on cyclic permutations

Rupert Li, Massachusetts Institute of Technology
2021 Enumerative Combinatorics and Applications
unavoidable sets and maximal avoidable sets.  ...  In particular, we enumerate many avoidance classes of sets of vincular patterns of length 3, including a complete enumeration for all single patterns of length 3.  ...  Joe Gallian for his editing feedback and operation of the Duluth REU program.  ...
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