Document Zbl 1548.92031

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Comput. Syst. Sci.</a> <a href="/?q=in%3A481956" title="Articles in this Issue">128, 1-17 (2022)</a>. </div> <div class="abstract">Summary: In the study of rooted phylogenetic networks, analyzing the set of rooted phylogenetic trees that are embedded in such a network is a recurring task. Algorithmically, this analysis requires a search of the exponential-sized multiset $S$ of rooted phylogenetic trees that are embedded in a rooted phylogenetic network $\mathcal{N} $. In this paper, we introduce the notion of a non-essential arc, which is an arc whose deletion from $\mathcal{N}$ results in a rooted phylogenetic network $\mathcal{N}^\prime$ that embeds the same set of rooted phylogenetic trees as $\mathcal{N}$ but whose associated multiset is smaller than $S$. We characterize non-essential arcs for the popular class of tree-child networks and show that identifying and deleting such arcs takes time that is cubic in the number of leaves of the network. Moreover, we show that deciding if a given arc of an arbitrary phylogenetic network is non-essential is $\Pi_2^P$-complete.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7526468">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D15" title="MSC2020">92D15</a> </td> <td class="space"> Problems related to evolution </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acaterpillar+ladder">caterpillar ladder</a>; <a href="/?q=ut%3Adisplaying+trees">displaying trees</a>; <a href="/?q=ut%3Anon-essential+arc">non-essential arc</a>; <a href="/?q=ut%3Aphylogenetic+network">phylogenetic network</a>; <a href="/?q=ut%3Atree-child">tree-child</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1548.92031" data-ciurl="/ci/07526468" data-biburl="/bibtex/07526468.bib" data-amsurl="/amsrefs/07526468.bib" data-xmlurl="/xml/07526468.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07526468.pdf" title="Zbl 1548.92031 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jcss.2022.02.005" aria-label="DOI for “Non-essential arcs in phylogenetic networks”" title="10.1016/j.jcss.2022.02.005">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2107.07721"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Anaya, M.; Anipchenko-Ulaj, O.; Ashfaq, A.; Chiu, J.; Kaiser, M.; Ohsawa, M. S.; Owen, M.; Pavlechko, E.; John, K. St.; Suleria, S.; Thompson, K.; Yap, C., On determining if tree-based networks contain fixed trees, Bull. Math. Biol., 78, 961-969 (2016) · <a href="/1348.92114" class="nowrap">Zbl 1348.92114</a></td> </tr><tr> <td>[2]</td> <td class="space">Bordewich, M.; Semple, C., Reticulation-visible networks, Adv. Appl. Math., 78, 114-141 (2016) · <a href="/1335.05170" class="nowrap">Zbl 1335.05170</a></td> </tr><tr> <td>[3]</td> <td class="space">Bentert, M.; Malik, J.; Weller, M., Tree containment with soft polytomies, (Scandinavian Symposium and Workshops on Algorithm Theory (2018)), 9:1-9:14 · <a href="/1477.68199" class="nowrap">Zbl 1477.68199</a></td> </tr><tr> <td>[4]</td> <td class="space">Cardona, G.; Rossello, F.; Valiente, G., Comparison of tree-child phylogenetic networks, IEEE/ACM Trans. Comput. Biol. Bioinform., 6, 552-569 (2009)</td> </tr><tr> <td>[5]</td> <td class="space">Cordue, P.; Linz, S.; Semple, C., Phylogenetic networks that display a tree twice, Bull. Math. Biol., 76, 2664-2679 (2014) · <a href="/1330.92085" class="nowrap">Zbl 1330.92085</a></td> </tr><tr> <td>[6]</td> <td class="space">Döcker, J.; Linz, S.; Semple, C., Display sets of normal and tree-child networks, Electron. J. Comb., 28 (2021), Paper 1.8 · <a href="/1456.05154" class="nowrap">Zbl 1456.05154</a></td> </tr><tr> <td>[7]</td> <td class="space">Cormen, T. H.; Leiserson, C. E.; Rivest, R. L.; Stein, C., Introduction to Algorithms (2001), MIT Press and McGraw-Hill: MIT Press and McGraw-Hill New York · <a href="/1047.68161" class="nowrap">Zbl 1047.68161</a></td> </tr><tr> <td>[8]</td> <td class="space">Döcker, J.; Linz, S.; Semple, C., Displaying trees across two phylogenetic networks, Theor. Comput. Sci., 796, 129-146 (2019) · <a href="/1435.68231" class="nowrap">Zbl 1435.68231</a></td> </tr><tr> <td>[9]</td> <td class="space">Felsenstein, J., Inferring Phylogenies (2004), Sinauer Associates</td> </tr><tr> <td>[10]</td> <td class="space">Fischer, M.; van Iersel, L.; Kelk, S.; Scornavacca, C., On computing the maximum parsimony score of a phylogenetic network, SIAM J. Discrete Math., 29, 559-585 (2015) · <a href="/1328.68300" class="nowrap">Zbl 1328.68300</a></td> </tr><tr> <td>[11]</td> <td class="space">Gambette, P.; Gunawan, A. D.M.; Labarre, A.; Vialette, S.; Zhang, L., Solving the tree containment problem in linear time for nearly stable phylogenetic networks, Discrete Appl. Math., 246, 62-79 (2018) · <a href="/1390.05083" class="nowrap">Zbl 1390.05083</a></td> </tr><tr> <td>[12]</td> <td class="space">Garey, M. R.; Johnson, D. S., Computers and Intractability: a Guide to the Theory of NP-Completeness (1979), W. H. Freeman and Company · <a href="/0411.68039" class="nowrap">Zbl 0411.68039</a></td> </tr><tr> <td>[13]</td> <td class="space">Gusfield, D., Recombinatorics: The Algorithmics of Ancestral Recombination Graphs and Explicit Phylogenetic Networks (2014), MIT Press · <a href="/1316.92001" class="nowrap">Zbl 1316.92001</a></td> </tr><tr> <td>[14]</td> <td class="space">Hein, J., Reconstructing evolution of sequences subject to recombination using parsimony, Math. Biosci., 98, 185-200 (1990) · <a href="/0692.92014" class="nowrap">Zbl 0692.92014</a></td> </tr><tr> <td>[15]</td> <td class="space">Huson, D. H.; Rupp, R.; Scornavacca, C., Phylogenetic Networks: Concepts, Algorithms and Applications (2010), Cambridge University Press</td> </tr><tr> <td>[16]</td> <td class="space">van Iersel, L.; Kelk, S.; Stamoulis, G.; Stougie, L.; Boes, O., On unrooted and root-uncertain variants of several well-known phylogenetic network problems, Algorithmica, 80, 2993-3022 (2018) · <a href="/1410.68178" class="nowrap">Zbl 1410.68178</a></td> </tr><tr> <td>[17]</td> <td class="space">van Iersel, L.; Semple, C.; Steel, M., Locating a tree in a phylogenetic network, Inf. Process. Lett., 110, 1037-1043 (2010) · <a href="/1379.68184" class="nowrap">Zbl 1379.68184</a></td> </tr><tr> <td>[18]</td> <td class="space">Jin, G.; Nakhleh, L.; Snir, S.; Tuller, T., Maximum likelihood of phylogenetic networks, Bioinformatics, 22, 2604-2611 (2006)</td> </tr><tr> <td>[19]</td> <td class="space">Jin, G.; Nakhleh, L.; Snir, S.; Tuller, T., Parsimony score of phylogenetic networks: hardness results and a linear-time heuristic, IEEE/ACM Trans. Comput. Biol. Bioinform., 6, 495-505 (2009)</td> </tr><tr> <td>[20]</td> <td class="space">Kanj, I. A.; Nakhleh, L.; Than, C.; Xia, G., Seeing the trees and their branches in the network is hard, Theor. Comput. Sci., 401, 153-164 (2008) · <a href="/1147.68059" class="nowrap">Zbl 1147.68059</a></td> </tr><tr> <td>[21]</td> <td class="space">Linz, S.; John, K. St.; Semple, C., Counting trees in a phylogenetic network is #P-complete, SIAM J. Comput., 42, 1768-1776 (2013) · <a href="/1276.05028" class="nowrap">Zbl 1276.05028</a></td> </tr><tr> <td>[22]</td> <td class="space">McDiarmid, C.; Semple, C.; Welsh, D., Counting phylogenetic networks, Ann. Comb., 19, 205-224 (2015) · <a href="/1310.05120" class="nowrap">Zbl 1310.05120</a></td> </tr><tr> <td>[23]</td> <td class="space">Nakhleh, L.; Jin, G.; Zhao, F.; Mellor-Crummey, J., Reconstructing phylogenetic networks using maximum parsimony, (IEEE Computational Systems Bioinformatics Conference (2005)), 93-102</td> </tr><tr> <td>[24]</td> <td class="space">Nguyen, C. T.; Nguyen, N. B.; Sung, W.-K.; Zhang, L., Reconstructing recombination network from sequence data: the small parsimony problem, IEEE/ACM Trans. Comput. Biol. Bioinform., 4, 394-402 (2007)</td> </tr><tr> <td>[25]</td> <td class="space">Semple, C., Phylogenetic networks with every embedded phylogenetic tree a base tree, Bull. Math. Biol., 78, 132-137 (2016) · <a href="/1356.92067" class="nowrap">Zbl 1356.92067</a></td> </tr><tr> <td>[26]</td> <td class="space">Stockmeyer, L. J., The polynomial-time hierarchy, Theor. Comput. Sci., 3, 1-22 (1976) · <a href="/0353.02024" class="nowrap">Zbl 0353.02024</a></td> </tr><tr> <td>[27]</td> <td class="space">Willson, S. J., Tree-average distances on certain phylogenetic networks have their weights uniquely determined, Algorithms Mol. Biol., 7, 13 (2012)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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