{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:34:03Z","timestamp":1753882443335,"version":"3.41.2"},"reference-count":5,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2022,7]]},"abstract":"<jats:p> Let [Formula: see text] be a function on a graph [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is said to be undefended with respect to [Formula: see text] if it is not adjacent to a vertex [Formula: see text] with [Formula: see text]. A function [Formula: see text] is called a weak Roman dominating function (WRDF) if each vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text], such that the function [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text], has no undefended vertex. The weight of a WRDF is the sum of its function values over all vertices, and the weak Roman domination number [Formula: see text] is the minimum weight of a WRDF in [Formula: see text]. In this paper, we consider the effects of edge deletion on the weak Roman domination number of a graph. We show that the deletion of an edge of [Formula: see text] can increase the weak Roman domination number by at most 1. Then we give a necessary condition for [Formula: see text]-ER-critical graphs, that is, graphs [Formula: see text] whose weak Roman domination number increases by the deletion of any edge. Restricted to the class of trees, we provide a constructive characterization of all [Formula: see text]-ER-critical trees. <\/jats:p>","DOI":"10.1142\/s1793830922500069","type":"journal-article","created":{"date-parts":[[2021,11,12]],"date-time":"2021-11-12T06:52:25Z","timestamp":1636699945000},"source":"Crossref","is-referenced-by-count":0,"title":["Graphs whose weak Roman domination number increases by the deletion of any edge"],"prefix":"10.1142","volume":"14","author":[{"given":"Rihab","family":"Hamid","sequence":"first","affiliation":[{"name":"LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nour","family":"El Houda Bendahib","sequence":"additional","affiliation":[{"name":"LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mustapha","family":"Chellali","sequence":"additional","affiliation":[{"name":"LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nac\u00e9ra","family":"Meddah","sequence":"additional","affiliation":[{"name":"LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2021,11,11]]},"reference":[{"key":"S1793830922500069BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-51117-3_11"},{"key":"S1793830922500069BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-58892-2_10"},{"key":"S1793830922500069BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.akcej.2019.12.001"},{"key":"S1793830922500069BIB004","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2003.06.004"},{"key":"S1793830922500069BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(02)00811-7"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830922500069","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,2]],"date-time":"2022-08-02T04:52:43Z","timestamp":1659415963000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S1793830922500069"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,11]]},"references-count":5,"journal-issue":{"issue":"05","published-print":{"date-parts":[[2022,7]]}},"alternative-id":["10.1142\/S1793830922500069"],"URL":"https:\/\/doi.org\/10.1142\/s1793830922500069","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"type":"print","value":"1793-8309"},{"type":"electronic","value":"1793-8317"}],"subject":[],"published":{"date-parts":[[2021,11,11]]},"article-number":"2250006"}}