|
==History==
According to <ref name="KL15">[https://www.springer.com/la/book/9783319258577 S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015.] {{ISBN|978-3-319-25859-1}}</ref>, Sergey Kitaev introduced the theory of word-representable graphs in 2004 based on joint research with Steven Seif <ref name="KS08">S. Kitaev and S. Seif. Word problem of the Perkins semigroup via directed acyclic graphs, Order 25 (2008), 177−194.</ref> on the ''Perkins semigroup'', which has played an important role in semigroup theory since 1960. However, the first systematic study of word-representable graphs was not undertaken until the 2008 paper <ref name="KP08">[https://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/repgr.pdf S. Kitaev and A. Pyatkin. On representable graphs, J. Autom., Lang. and Combin. 13 (2008), 45−54.]</ref> by Sergey Kitaev and Artem Pyatkin starting development of the theory. One of key contributors to the area is [http://www.ru.is/~mmh/ Magnús M. Halldórsson] <ref name=":0">[https://link.springer.com/chapter/10.1007/978-3-642-14455-4_41 M.M. Halldórsson, S. Kitaev, A. Pyatkin (2010) Graphs capturing alternations in words. In: Y. Gao, H. Lu, S. Seki, S. Yu (eds), Developments in Language Theory. DLT 2010. Lecture Notes Comp. Sci. 6224, Springer, 436−437.]</ref> <ref name=":1">[https://link.springer.com/chapter/10.1007/978-3-642-25870-1_18 M.M. Halldórsson, S. Kitaev, A. Pyatkin (2011) Alternation graphs. In: P. Kolman, J. Kratochvíl (eds), Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes Comp. Sci. 6986, Springer, 191−202.]</ref> <ref name="HKP16">[https://www.sciencedirect.com/science/article/pii/S0166218X15003868 M. Halldórsson, S. Kitaev and A. Pyatkin. Semi-transitive orientations and word-representable graphs, Discr. Appl. Math. 201 (2016), 164−171.]</ref>. Up to date, 35+ papers have been written on the subject, and the core of the book <ref name="KL15" /> by Sergey Kitaev and Vadim Lozin is devoted to the theory of word-representable graphs. A quick way to get familiar with the area is to read one of the survey articles <ref name=":2">[[arxiv:1705.05924|S. Kitaev. A comprehensive introduction to the theory of word-representable graphs.]] In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67.</ref> <ref>[https://link.springer.com/article/10.1134/S1990478918020084 S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296.]</ref> <ref>[http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53]</ref>.
==Motivation to study the graphs==
==Early results==
It was shown in <ref name="KP08" /> that a graph ''G'' is word-representable iff it is '''k-representable''' for some ''k'', that is, ''G'' can be represented by a word having ''k'' copies of each letter. Moreover, if a graph is ''k''-representable then it is also (''k'' + 1)-representable. Thus, the notion of the '''representation number of a graph''', as the ''minimum'' ''k'' such that a graph is word-representable, is well-defined. Non-word-representable graphs have the representation number ∞. Graphs with representation number 1 are precisely the set of [[Complete graph|complete graphs]], while graphs with representation number 2 are precisely the class of [[Circle graph|circle]] non-complete graphs. In particular, [http://mathworld.wolfram.com/Forest.html forests] (except for single [[Tree (graph theory)|trees]] on at most 2 vertices), [[Ladder graph|ladder graphs]] and [[Cycle graph|cycle graphs]] have representation number 2. No classification for graphs with representation number 3 is known. However, there are examples of such graphs, e.g. [[Petersen graph|Petersen's graph]] and [http://mathworld.wolfram.com/PrismGraph.html prisms]. Moreover, 3-[[Homeomorphism (graph theory)|subdivision]] of any graph is 3-representable. In particular, for every graph ''G'' there exists a 3-representable graph ''H'' that contains ''G'' as a [[Graph minor|minor]] <ref name="KP08" />.
A graph ''G'' is '''permutationally representable''' if it can be represented by a word of the form ''p''<sub>1</sub>''p''<sub>2</sub>...''p<sub>k</sub>'', where ''p<sub>i</sub>'' is a [[permutation]]. On can also say that ''G'' is '''permutationally ''k''-representable'''. A graph is permutationally representable iff it is a [[Comparability graph|comparability graph]] <ref name="KS08" />. A graph is word-representable implies that the [[Neighbourhood (graph theory)|neighbourhood]] of each vertex is permutationally representable (i.e. is a [[comparability graph]]) <ref name="KS08" />. Converse to the last statement is not true <ref name=":0" />. However, the fact that the [[Neighbourhood (graph theory)|neighbourhood]] of each vertex is a [[comparability graph]] implies that the [[Clique problem|Maximum Clique problem]] is polynomially solvable on word-representable graphs <ref name=":1" /> <ref name="HKP16" />.
==Semi-transitive orientations==
Semi-transitive orientations provide a powerful tool to study word-representable graphs. A directed graph is '''semi-transitively oriented''' iff it is [http://mathworld.wolfram.com/AcyclicGraph.html acyclic] and for any directed path ''u''<sub>1</sub>→''u''<sub>2</sub>→ ...→''u<sub>t</sub>'', ''t'' ≥ 2, either there is no edge from ''u''<sub>1</sub> to ''u<sub>t</sub>'' or all edges ''u<sub>i</sub>'' → ''u<sub>j</sub>'' exist for 1 ≤ ''i'' < ''j'' ≤ ''t''. A key theorem in the theory of word-representable graphs states that a graph is word-representable iff it admits a semi-transitive orientation <ref name="HKP16" />. As a corollary to the proof of the key theorem one obtain an upper bound on word-representants: Each non-complete word-representable graph ''G'' is 2(''n'' − ''κ''(''G''))-representable, where ''κ''(''G'') is the size of a maximal clique in ''G'' <ref name="HKP16" />. As an immediate corollary of the last statement, one has that the [https://www.encyclopediaofmath.org/index.php/Recognition_problem recognition problem] of word-representability is in [[NP (complexity)|NP]]. In 2014, [http://fc.isima.fr/~limouzy/ Vincent Limouzy] observed that it is an [[NP-completeness|NP-complete problem]] to recognise whether a given graph is word-representable <ref name="KL15" /> <ref name=":2" />. Another important corollary to the key theorem is that any [[Graph coloring|3-colorable<sub>2''n''+1</sub>, graph]]for ''n'' ≥ 2, are not word-representable and ''W''<sub>5</sub> is the minimum (by the number of vertices) non-word-representable graph. TheTaking lastany factnon-comparability impliesgraph thatand adding an apex (a vertex connected to any other vertex), we obtain a non-word-representable graph, which then can produce infinitely many classicalnon-word-representable graphs <ref name="KL15" />. Any graph problemsproduced arein NPthis way will necessarily have a triangle (a cycle of length 3), and a vertex of degree at least 5. Non-hardword-representable graphs of maximum degree 4 exist <ref>[[arxiv:1307.1810|A. Collins, S. Kitaev, and V. Lozin. New results on word-representable graphs, Discr. Appl. Math. 216 (2017), 136–141.]]</ref> and non-word-representable triangle-free graphs exist <ref name=":1" />. Regular non-word representable graphs also exist <ref name="KL15" />. Non-isomorphic non-word-representable connected graphs on at most eight vertices were first enumerated by Heman Z.Q. Chen. His calculations were extended in <ref>https://cs.uwaterloo.ca/journals/JIS/VOL22/Kitaev/kitaev11.html Ö. Akgün, I.P. Gent, S. Kitaev, H. Zantema. Solving computational problems in the theory of word-representable graphs. Journal of Integer Sequences 22 (2019), Article</ref>, where it was shown that the numbers of non-isomorphic non-word-representable connected graphs on 5−11 vertices are given, respectively, by 0, 1, 25, 929, 54957, 4880093, 650856040. This is the sequence A290814 in the http://oeis.org/search?q=1%2C+25%2C+929%2C+54957%2C+4880093&language=english&go=Search Online Encyclopaedia of Integer Sequences<ref name="KL15" />. The operations not necessarily preserving word-representability are taking the complement, taking the line graph, edge contraction <ref name="KL15" />, gluing two graphs in a clique of size 2 or more <ref name="CKS2019" />, tensor product, lexicographic product and strong product <ref name="CKK2019">[https://link.springer.com/content/pdf/10.1007/s10878-018-0358-7.pdf I. Choi, J. Kim, and M. Kim. On operations preserving semi-transitive orient ability of graphs, Journal of Combinatorial Optimization 37 (2019) 4, 1351−1366.]</ref>. Edge-deletion, edge-addition and edge-lifting with respect to word-representability (equivalently, semi-transitive orientability) are studied in <ref name="CKK2019" />.
==Overview of selected results==
===Non-word-representable graphs===
[[Wheel graph|Wheel graphs]] ''W''<sub>2''n''+1</sub>, for ''n'' ≥ 2, are not word-representable and ''W''<sub>5</sub> is the minimum (by the number of vertices) non-word-representable graph. Taking any non-comparability graph and adding an apex (a vertex connected to any other vertex), we obtain a non-word-representable graph, which then can produce infinitely many non-word-representable graphs <ref name="KL15" />. Any graph produced in this way will necessarily have a triangle (a cycle of length 3), and a vertex of degree at least 5. Non-word-representable graphs of maximum degree 4 exist <ref>[[arxiv:1307.1810|A. Collins, S. Kitaev, and V. Lozin. New results on word-representable graphs, Discr. Appl. Math. 216 (2017), 136–141.]]</ref> and non-word-representable triangle-free graphs exist <ref name=":1" />. Regular non-word representable graphs also exist <ref name="KL15" />. Non-isomorphic non-word-representable connected graphs on at most eight vertices were first enumerated by Heman Z.Q. Chen. His calculations were extended in <ref>[https://cs.uwaterloo.ca/journals/JIS/VOL22/Kitaev/kitaev11.html Ö. Akgün, I.P. Gent, S. Kitaev, H. Zantema. Solving computational problems in the theory of word-representable graphs. Journal of Integer Sequences 22 (2019), Article 19.2.5.]
</ref>, where it was shown that the numbers of non-isomorphic non-word-representable connected graphs on 5−11 vertices are given, respectively, by 0, 1, 25, 929, 54957, 4880093, 650856040. This is the sequence A290814 in the [http://oeis.org/search?q=1%2C+25%2C+929%2C+54957%2C+4880093&language=english&go=Search Online Encyclopaedia of Integer Sequences (OEIS)].
===Operations on graphs and word-representability===
Operations preserving word-representability are removing a vertex, replacing a vertex with a module, Cartesian product, rooted product, subdivision of a graph, connecting two graphs by an edge and gluing two graphs in a vertex <ref name="KL15" />. The operations not necessarily preserving word-representability are taking the complement, taking the line graph, edge contraction <ref name="KL15" />, gluing two graphs in a clique of size 2 or more <ref name="CKS2019" />, tensor product, lexicographic product and strong product <ref name="CKK2019">[https://link.springer.com/content/pdf/10.1007/s10878-018-0358-7.pdf I. Choi, J. Kim, and M. Kim. On operations preserving semi-transitive orient ability of graphs, Journal of Combinatorial Optimization 37 (2019) 4, 1351−1366.]</ref>. Edge-deletion, edge-addition and edge-lifting with respect to word-representability (equivalently, semi-transitive orientability) are studied in <ref name="CKK2019" />.
===Graphs with high representation number===
|NP-hard
|-
|[https://cs.stanford.edu/people/eroberts/courses/soco/projects/dna-computing/clique.htm Maximum<sup>1−''ε''</sup> for any ''ε'' > Clique]0
|in P
|-
|approximating the graph representation number within a factor ''n''<sup>1−''ε''</sup> for any ''ε'' > 0
|NP-hard
|}
==Generalisations==
A number of generalisations <ref name="JKPR2015">[https://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p53 M. Jones, S. Kitaev, A. Pyatkin, and J. Remmel. Representing Graphs via Pattern Avoiding Words, Electron. J. Combin. 22 (2), Res. Pap. P2.53, 1−20 (2015).]</ref> <ref name="GJ2019">[https://www.lehmanns.de/shop/mathematik-informatik/48968161-9783030287955-combinatorics-on-words M. Gaetz and C. Ji. Enumeration and extensions of word-representable graphs. Lecture Notes in Computer Science 11682 (2019) 180−192. In R. Mercas, D. Reidenbach (Eds) Combinatorics on Words. WORDS 2019.]</ref> <ref name="GJ2019-2">[[arXiv:1909.00019|M. Gaetz and C. Ji. Enumeration and Extensions of Word-representants, arXiv:1909.00019.]]</ref> of the notion of a word-representable graph are based on the observation by [https://www.math.ucsd.edu/memorials/jeffrey-remmel/ Jeff Remmel] that non-edges are defined by occurrences of the pattern 11 (two consecutive equal letters) in a word representing a graph, while edges are defined by avoidance of this pattern. For example, instead of the pattern 11, one can use the pattern 112, or the pattern, 1212, or any other binary pattern where the assumption that the alphabet is ordered can be made so that a letter in a word corresponding to 1 in the pattern is less than that corresponding to 2 in the pattern. Letting ''u'' be an ordered binary pattern we thus get the notion of a '''''u''-representable graph'''. So, word-representable graphs are just the class of 11-representable graphs. Intriguingly, '''any graph''' can be ''u''-represented assuming ''u'' is of length at least 3 <ref name="K2017">[https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.22097 S. Kitaev. Existence of u-representation of graphs, Journal of Graph Theory 85 (2017) 3, 661−668.]</ref>.
Another way to generalise the notion of a word-representable graph, again suggested by [https://www.math.ucsd.edu/memorials/jeffrey-remmel/ Jeff Remmel], is to introduce the "degree of tolerance" ''k'' for occurrences of a pattern ''p'' defining edges/non-edges. That is, we can say that if there are up to ''k'' occurrence of ''p'' formed by letters ''a'' and ''b'', then there is an edge between ''a'' and ''b''. This gives the notion of a ''k''-''p''-representable graph, and ''k''-11-representable graphs are studied in <ref name="CKKK2019">[[arxiv:1803.01055|G.-S. Cheon, J. Kim, M. Kim, and A. Pyatkin. On ''k''-11-representable graphs. J. Combin. 10 (2019) 3, 491−513.]]</ref>. Note that 0-11-representable graphs are precisely word-representable graphs. The key results in <ref name="CKKK2019" /> are that '''any''' graph is 2-11-representable and that word-representable graphs are a proper subclass of 1-11-representable graphs. Whether or not any graph is 1-11-representable is a challenging open problem.
For yet another type of relevant generalisation, [[Hans Zantema]] suggested the notion of a '''''k''-semi-transitive orientation''' refining the notion of a semi-transitive orientation <ref name="AGKZ2019" />. The idea here is restricting ourselves to considering ''only'' directed paths of length not exceeding ''k'' while allowing violations of semi-transitivity on longer paths.
The list of publications to study representation of graphs by words contains, but is not limited to
# [https://cs.uwaterloo.ca/journals/JIS/VOL22/Kitaev/kitaev11.pdf Ö. Akgün, I.P. Gent, S. Kitaev, H. Zantema. Solving computational problems in the theory of word-representable graphs. Journal of Integer Sequences 22 (2019), Article<!-- 19Inline citations added to your article will automatically display here.2 See https://en.5wikipedia.]org/wiki/WP:REFB for instructions on how to add citations. -->
# [[arxiv:1405.3527|P. Akrobotu, S. Kitaev, and Z. Masárová. On word-representability of polyomino triangulations. Siberian Adv. Math. 25 (2015), 1−10.]]
# B. Broere. Word representable graphs, 2018. Master thesis at Radboud University, Nijmegen.
# [[arxiv:1808.01800|B. Broere and H. Zantema. "The ''k''-cube is ''k''-representable," J. Autom., Lang., and Combin. 24 (2019) 1, 3-12.]]
# [[arXiv:1911.00408|J. N. Chen and S. Kitaev. On the 12-representability of induced subgraphs of a grid graph, Discussiones Mathematicae Graph Theory, to appear]]
# [[arXiv:1909.09471|T. Z. Q. Chen, S. Kitaev, and A. Saito. Representing split graphs by words, arXiv:1909.09471]]
# [[arxiv:1503.08002|T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of face subdivisions of triangular grid graphs, Graphs and Combin. 32(5) (2016), 1749−1761.]]
# [[arxiv:1507.06749|T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of triangulations of grid-covered cylinder graphs, Discr. Appl. Math. 213 (2016), 60−70.]]
# [[arXiv:1907.09152|G.-S. Cheon, J. Kim, M. Kim, and S. Kitaev. Word-representability of Toeplitz graphs, Discr. Appl. Math., to appear.]]
# [[arxiv:1803.01055|G.-S. Cheon, J. Kim, M. Kim, and A. Pyatkin. On ''k''-11-representable graphs. J. Combin. 10 (2019) 3, 491−513.]]
# [https://link.springer.com/content/pdf/10.1007/s10878-018-0358-7.pdf I. Choi, J. Kim, and M. Kim. On operations preserving semi-transitive orient ability of graphs, Journal of Combinatorial Optimization 37 (2019) 4, 1351−1366.]
# [[arxiv:1307.1810|A. Collins, S. Kitaev, and V. Lozin. New results on word-representable graphs, Discr. Appl. Math. 216 (2017), 136−141.]]
# [[arxiv:1806.04673|A. Daigavane, M. Singh, B.K. George. 2-uniform words: cycle graphs, and an algorithm to verify specific word-representations of graphs. arXiv:1806.04673 (2018).]]
# [https://www.lehmanns.de/shop/mathematik-informatik/48968161-9783030287955-combinatorics-on-words M. Gaetz and C. Ji. Enumeration and extensions of word-representable graphs. Lecture Notes in Computer Science 11682 (2019) 180−192. In R. Mercas, D. Reidenbach (Eds) Combinatorics on Words. WORDS 2019.]
# [[arXiv:1909.00019|M. Gaetz and C. Ji. Enumeration and Extensions of Word-representants, arXiv:1909.00019.]]
# [[arxiv:1602.08965|A. Gao, S. Kitaev, and P. Zhang. On 132-representable graphs. Australasian J. Combin. 69 (2017), 105−118.]]
# [[arxiv:1605.01688|M. Glen. Colourability and word-representability of near-triangulations, Pure and Applied Mathematics, to appear, 2019.]]
# M. Glen. On word-representability of polyomino triangulations & crown graphs, 2019. PhD thesis, University of Strathclyde.
# [[arxiv:1503.05076|M. Glen and S. Kitaev. Word-Representability of Triangulations of Rectangular Polyomino with a Single Domino Tile, J. Combin.Math. Combin. Comput. 100, 131−144, 2017.]]
# [https://www.sciencedirect.com/science/article/pii/S0166218X18301045 M. Glen, S. Kitaev, and A. Pyatkin. On the representation number of a crown graph, Discr. Appl. Math. 244, 2018, 89−93.]
# [https://pdfs.semanticscholar.org/a2df/a4c88505510ea1a7d4357972d9ab24575195.pdf M.M. Halldórsson, S. Kitaev, A. Pyatkin (2010) Graphs capturing alternations in words. In: Y. Gao, H. Lu, S. Seki, S. Yu (eds), Developments in Language Theory. DLT 2010. Lecture Notes Comp. Sci. 6224, Springer, 436−437.]
# [https://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/wg2.pdf M.M. Halldórsson, S. Kitaev, A. Pyatkin (2011) Alternation graphs. In: P. Kolman, J. Kratochvíl (eds), Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes Comp. Sci. 6986, Springer, 191−202.]
# [[arxiv:1501.07108|M. Halldórsson, S. Kitaev and A. Pyatkin. Semi-transitive orientations and word-representable graphs, Discr. Appl. Math. 201 (2016), 164−171.]]
# [https://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p53 M. Jones, S. Kitaev, A. Pyatkin, and J. Remmel. Representing Graphs via Pattern Avoiding Words, Electron. J. Combin. 22 (2), Res. Pap. P2.53, 1−20 (2015).]
# [[arxiv:1403.1616|S. Kitaev. On graphs with representation number 3, J. Autom., Lang. and Combin. 18 (2013), 97−112.]]
# [[arxiv:1705.05924|S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67.]]
# [https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.22097 S. Kitaev. Existence of u-representation of graphs, Journal of Graph Theory 85 (2017) 3, 661−668.]
# [[arxiv:1709.09725|S. Kitaev, Y. Long, J. Ma, H. Wu. Word-representability of split graphs, arXiv:1709.09725 (2017).]]
# [[Special:BookSources/978-3-319-25859-1|S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015. ISBN 978-3-319-25859-1]].
# [https://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/repgr.pdf S. Kitaev and A. Pyatkin. On representable graphs, J. Autom., Lang. and Combin. 13 (2008), 45−54.]
# [https://link.springer.com/article/10.1134%2FS1990478918020084 S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296.]
#[[arxiv:1903.02777|S. Kitaev and A. Saito. On semi-transitive orientability of Kneser graphs and their complements, arxiv:1903.02777.]]
# [http://120.52.51.19/file.scirp.org/pdf/OJDM20110200008_15101466.pdf S. Kitaev, P. Salimov, C. Severs, and H. Úlfarsson. Word-representability of line graphs, Open Journal of Discrete Mathematics 1 (2011) 2, 96−101.]
# [[arxiv:1102.3980|S. Kitaev, P. Salimov, C. Severs, and H. Úlfarsson (2011) On the representability of line graphs. In: G. Mauri, A. Leporati (eds), Developments in Language Theory. DLT 2011. Lecture Notes Comp. Sci. 6795, Springer, 478−479.]]
# [https://link.springer.com/article/10.1007/s11083-008-9083-7 S. Kitaev and S. Seif. Word problem of the Perkins semigroup via directed acyclic graphs, Order 25 (2008), 177−194.]
# [https://matheo.uliege.be/bitstream/2268.2/6988/4/Memoire_Leloup.pdf E. Leloup. Graphes représentables par mot. Master Thesis, University of Liège, 2019]
# [https://www.dmgt.uz.zgora.pl/publish/view_pdf.php?ID=-2009|Y. Mandelshtam. On graphs representable by pattern-avoiding words, Discussiones Mathematicae Graph Theory 39 (2019) 375−389.]
# [https://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/wrg-obzor.pdf С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53.]
== Software ==
Software to study word-representable graphs can be found here:
#[http://personal.strath.ac.uk/sergey.kitaev/word-representable-graphs.html M. Glen. Software to deal with word-representable graphs, 2017. Available at https://personal.cis.strath.ac.uk/sergey.kitaev/word-representable-graphs.html.]
# [https://www.win.tue.nl/~hzantema/reprnr.html H. Zantema. Software REPRNR to compute the representation number of a graph, 2018. Available at https://www.win.tue.nl/~hzantema/reprnr.html.]
== References ==
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{{reflist}}
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