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However, checking that a model satisfies the adhesivity properties is sometimes far from immediate. In this paper we present a new criterion giving a sufficient condition for<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {M},\\mathcal {N}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>M<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>N<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula><jats:italic>-adhesivity<\/jats:italic>, a generalisation of the original notion of adhesivity. We apply it to several existing categories, and in particular to<jats:italic>hierarchical graphs<\/jats:italic>, a formalism that is notoriously difficult to fit in the mould of algebraic approaches to rewriting and for which various alternative definitions float around.<\/jats:p>","DOI":"10.1007\/978-3-030-99253-8_11","type":"book-chapter","created":{"date-parts":[[2022,3,28]],"date-time":"2022-03-28T20:02:48Z","timestamp":1648497768000},"page":"205-224","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A new criterion for $$\\mathcal {M}, \\mathcal {N}$$-adhesivity, with an application to hierarchical graphs"],"prefix":"10.1007","author":[{"given":"Davide","family":"Castelnovo","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0690-3051","authenticated-orcid":false,"given":"Fabio","family":"Gadducci","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0755-3444","authenticated-orcid":false,"given":"Marino","family":"Miculan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,29]]},"reference":[{"key":"11_CR1","unstructured":"J. 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