{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:11:06Z","timestamp":1759335066836,"version":"build-2065373602"},"reference-count":19,"publisher":"World Scientific Pub Co Pte Ltd","issue":"06","funder":[{"name":"Qinghai Key Laboratory of Internet of Things Project","award":["2017-ZJ-Y21"],"award-info":[{"award-number":["2017-ZJ-Y21"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Found. Comput. Sci."],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:p> Let G be a graph and [Formula: see text]. The vertex w is said to resolve a pair u and v if and only if [Formula: see text]. A set [Formula: see text] is defined as a resolving set of G if for all [Formula: see text], the pair [Formula: see text] is resolved by some [Formula: see text]. The minimum cardinality of a resolve set of G is defined as [Formula: see text]. A set [Formula: see text] is a local resolve set of G if for all [Formula: see text] such that [Formula: see text], the pair [Formula: see text] is resolved by some [Formula: see text]. The minimum cardinality of a local resolve set of G is defined as [Formula: see text]. An edge [Formula: see text] is said to be monitored by [Formula: see text] if [Formula: see text] or [Formula: see text]. A set [Formula: see text] is a distance-edge-monitoring (DEM) set if for all [Formula: see text], e is monitored by some [Formula: see text]. The minimum cardinality of a [Formula: see text] set of G is defined as [Formula: see text]. <\/jats:p><jats:p> In this paper, we obtained that [Formula: see text] for all non trivial graphs with order n, and the exact value of [Formula: see text] and [Formula: see text] for [Formula: see text], [Formula: see text]. Also, we obtained that if [Formula: see text] then [Formula: see text]. With respect to the relation between the defined graph invariants, it was proved a bound for [Formula: see text] for [Formula: see text] and the exact values of [Formula: see text] for [Formula: see text], where [Formula: see text] (resp, [Formula: see text])is the maximum value of [Formula: see text] (resp, [Formula: see text]) over all graphs G with order n. Finally, we proved that for [Formula: see text], there exists a graph with order n such that [Formula: see text] and [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0129054124500230","type":"journal-article","created":{"date-parts":[[2024,11,7]],"date-time":"2024-11-07T09:01:05Z","timestamp":1730970065000},"page":"901-920","source":"Crossref","is-referenced-by-count":0,"title":["The Local Metric Dimension and Distance-Edge-Monitoring Number of Graph"],"prefix":"10.1142","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2523-7090","authenticated-orcid":false,"given":"Chenxu","family":"Yang","sequence":"first","affiliation":[{"name":"School of Science, Tianjin Chengjian University, Tianjin, Tianjin 300384, P. R. China"},{"name":"School of Computer, Qinghai Normal University, Xining, Qinghai 810008, P. R. 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