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{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,13]],"date-time":"2024-01-13T16:03:59Z","timestamp":1705161839081},"reference-count":74,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,8,12]],"date-time":"2023-08-12T00:00:00Z","timestamp":1691798400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,8,12]],"date-time":"2023-08-12T00:00:00Z","timestamp":1691798400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["MI439\/14-2"]},{"DOI":"10.13039\/501100009244","name":"Stockholm University","doi-asserted-by":"crossref"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Theory Biosci."],"published-print":{"date-parts":[[2023,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Rooted acyclic graphs appear naturally when the phylogenetic relationship of a set <jats:italic>X<\/jats:italic> of taxa involves not only speciations but also recombination, horizontal transfer, or hybridization that cannot be captured by trees. A variety of classes of such networks have been discussed in the literature, including phylogenetic, level-1, tree-child, tree-based, galled tree, regular, or normal networks as models of different types of evolutionary processes. Clusters arise in models of phylogeny as the sets <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\mathrm{\\texttt{C}}\\,}}(v)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mrow>\n <mml:mspace \/>\n <mml:mi>C<\/mml:mi>\n <mml:mspace \/>\n <\/mml:mrow>\n <mml:mo>(<\/mml:mo>\n <mml:mi>v<\/mml:mi>\n <mml:mo>)<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of descendant taxa of a vertex <jats:italic>v<\/jats:italic>. The clustering system <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {C}_N$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msub>\n <mml:mi>C<\/mml:mi>\n <mml:mi>N<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> comprising the clusters of a network <jats:italic>N<\/jats:italic> conveys key information on <jats:italic>N<\/jats:italic> itself. In the special case of rooted phylogenetic trees, <jats:italic>T<\/jats:italic> is uniquely determined by its clustering system <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {C}_T$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msub>\n <mml:mi>C<\/mml:mi>\n <mml:mi>T<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Although this is no longer true for networks in general, it is of interest to relate properties of <jats:italic>N<\/jats:italic> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {C}_N$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msub>\n <mml:mi>C<\/mml:mi>\n <mml:mi>N<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Here, we systematically investigate the relationships of several well-studied classes of networks and their clustering systems. The main results are correspondences of classes of networks and clustering systems of the following form: If <jats:italic>N<\/jats:italic> is a network of type <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {X}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mi>X<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {C}_N$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msub>\n <mml:mi>C<\/mml:mi>\n <mml:mi>N<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> satisfies <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {Y}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mi>Y<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and conversely if <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {C}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mi>C<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a clustering system satisfying <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {Y},$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mi>Y<\/mml:mi>\n <mml:mo>,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> then there is network <jats:italic>N<\/jats:italic> of type <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {X}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mi>X<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {C}\\subseteq \\mathscr {C}_N$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mi>C<\/mml:mi>\n <mml:mo>\u2286<\/mml:mo>\n <mml:msub>\n <mml:mi>C<\/mml:mi>\n <mml:mi>N<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.<\/jats:p>","DOI":"10.1007\/s12064-023-00398-w","type":"journal-article","created":{"date-parts":[[2023,8,12]],"date-time":"2023-08-12T18:02:01Z","timestamp":1691863321000},"page":"301-358","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Clustering systems of phylogenetic networks"],"prefix":"10.1007","volume":"142","author":[{"given":"Marc","family":"Hellmuth","sequence":"first","affiliation":[]},{"given":"David","family":"Schaller","sequence":"additional","affiliation":[]},{"given":"Peter F.","family":"Stadler","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,8,12]]},"reference":[{"key":"398_CR1","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1137\/0201008","volume":"1","author":"AV Aho","year":"1972","unstructured":"Aho AV, Garey MR, Ullman JD (1972) The transitive reduction of a directed graph. 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