{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T14:50:46Z","timestamp":1770994246737,"version":"3.50.1"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2019,1,31]],"date-time":"2019-01-31T00:00:00Z","timestamp":1548892800000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,1,31]],"date-time":"2019-01-31T00:00:00Z","timestamp":1548892800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,1,31]],"date-time":"2019-01-31T00:00:00Z","timestamp":1548892800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>The recent increase of interest in the graph invariant called tree-depth and in its applications in algorithms and logic on graphs led to a natural question: is there an analogously useful &amp;quot;depth&amp;quot; notion also for dense graphs (say; one which is stable under graph complementation)? To this end, in a 2012 conference paper, a new notion of shrub-depth has been introduced, such that it is related to the established notion of clique-width in a similar way as tree-depth is related to tree-width. Since then shrub-depth has been successfully used in several research papers. Here we provide an in-depth review of the definition and basic properties of shrub-depth, and we focus on its logical aspects which turned out to be most useful. In particular, we use shrub-depth to give a characterization of the lower ${\\omega}$ levels of the MSO1 transduction hierarchy of simple graphs.<\/jats:p>","DOI":"10.23638\/lmcs-15(1:7)2019","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:33:50Z","timestamp":1743701630000},"source":"Crossref","is-referenced-by-count":4,"title":["Shrub-depth: Capturing Height of Dense Graphs"],"prefix":"10.23638","volume":"Volume 15, Issue 1","author":[{"given":"Robert","family":"Ganian","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2125-1514","authenticated-orcid":false,"given":"Petr","family":"Hlin\u011bn\u00fd","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jaroslav","family":"Ne\u0161et\u0159il","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jan","family":"Obdr\u017e\u00e1lek","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0724-3729","authenticated-orcid":false,"given":"Patrice Ossona","family":"de Mendez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2019,1,31]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1707.00359v4","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1707.00359v4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:33:50Z","timestamp":1743701630000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/3798"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,31]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-15(1:7)2019","relation":{"has-preprint":[{"id-type":"arxiv","id":"1707.00359v3","asserted-by":"subject"},{"id-type":"arxiv","id":"1707.00359v2","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1707.00359","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1707.00359","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"value":"1860-5974","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1,31]]},"article-number":"3798"}}