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Two Element Unavoidable Sets of Partial Words
[chapter]

*
Lecture Notes in Computer Science
*

We pose a conjecture, and show that affirmative proof

doi:10.1007/978-3-540-73208-2_12
fatcat:ys2edvkemfebzlvyp3glw67oxu
*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]

2011
*
Lecture Notes in Computer Science
*

This is a step towards the difficult problem

doi:10.1007/978-3-642-19222-7_6
fatcat:2qnycuns55ge3eyy26vb55lnny
*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

2009
*
Theoretical Computer Science
*

We prove that the problem

doi:10.1016/j.tcs.2008.11.011
fatcat:24wzsl7blbbqbdxen3od4ushea
*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

2012
*
Journal of Discrete Algorithms
*

Towards this, we investigate the minimum size

doi:10.1016/j.jda.2011.12.002
fatcat:cwk24hdtr5e5zkjyejt4e4jgie
*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

doi:10.1201/9781420060935.ch12
fatcat:lnh37uk2xbbihdrqt7zrlbcx4u
*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

2008
*
Theory of Computing Systems
*

We pose a conjecture, and show that affirmative proof

doi:10.1007/s00224-008-9106-1
fatcat:3utlarf2cjeffbxf6p5jha6dda
*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

doi:10.1016/j.tcs.2010.09.006
fatcat:bwvs6vekt5a45lwze2xpxbs5ka
*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

doi:10.1007/978-3-642-02737-6_9
fatcat:agy5mp2yyjgvhb6yenw6yguj7a
*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

1998
*
Theoretical Computer Science
*

In this paper, I prove that the inventory

doi:10.1016/s0304-3975(97)00031-5
fatcat:k5h6xhddibditeyq5zn5cqzcdm
*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

1984
*
Discrete Applied Mathematics
*

A subset X

doi:10.1016/0166-218x(84)90014-3
fatcat:j5moxv2b3bhajkplswdirpc5we
*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

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

1997
*
Discrete Mathematics
*

We also use these results to show that the intricacy

doi:10.1016/s0012-365x(96)00354-8
fatcat:hs4s4s7hfvhk3kad7vleoxytwm
*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

2013
*
Journal of Discrete Algorithms
*

We complete the classification

doi:10.1016/j.jda.2013.06.007
fatcat:d45yfc2njvhnzkiqn7p47jggiy
*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

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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