{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:09:01Z","timestamp":1753880941694,"version":"3.41.2"},"reference-count":25,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2023,5]]},"abstract":"<jats:p>In a graph [Formula: see text], a module is a vertex subset [Formula: see text] of [Formula: see text] such that every vertex outside [Formula: see text] is adjacent to all or none of [Formula: see text]. For example, [Formula: see text], [Formula: see text][Formula: see text] and [Formula: see text] are modules of [Formula: see text], called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex [Formula: see text] of a prime graph [Formula: see text] is critical if [Formula: see text] is decomposable. Moreover, a prime graph with [Formula: see text] noncritical vertices is called [Formula: see text]-critical graph. A prime graph [Formula: see text] is [Formula: see text]-minimal if there is some [Formula: see text]-vertex set [Formula: see text] of vertices such that there is no proper induced subgraph of [Formula: see text] containing [Formula: see text] is prime. From this perspective, Boudabbous proposes to find the [Formula: see text]-critical graphs and [Formula: see text]-minimal graphs for some integer [Formula: see text] even in a particular case of graphs. This research paper attempts to answer Boudabbous\u2019s question. First, we describe the [Formula: see text]-critical tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-critical tree with [Formula: see text] vertices where [Formula: see text]. Second, we provide a complete characterization of the [Formula: see text]-minimal tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-minimal tree with [Formula: see text] vertices where [Formula: see text].<\/jats:p>","DOI":"10.1142\/s1793830922501099","type":"journal-article","created":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T09:43:54Z","timestamp":1651139034000},"source":"Crossref","is-referenced-by-count":0,"title":["Primality, criticality and minimality problems in trees"],"prefix":"10.1142","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6352-2138","authenticated-orcid":false,"given":"Walid","family":"Marweni","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Sfax University, Tunisia"}]}],"member":"219","published-online":{"date-parts":[[2022,5,24]]},"reference":[{"issue":"2","key":"S1793830922501099BIB001","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.ajmsc.2015.02.003","volume":"21","author":"Alzohairi M.","year":"2015","journal-title":"Arab J. 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