{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T07:07:42Z","timestamp":1777705662069,"version":"3.51.4"},"reference-count":13,"publisher":"SAGE Publications","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2023,5,4]]},"abstract":"<jats:p>In real-life scenarios, both the vertex weight and edge weight in a network are hard to define exactly. We can incorporate the fuzziness into a network to handle this type of uncertain situation. Here, we use triangular fuzzy number to describe the vertex weight and edge weight of a fuzzy network G. In this paper, we consider weighted k-domination problem in fuzzy network. The weighted k-domination (WKD) problem is to find a k dominating set D which minimizes the cost f\u00a0(D)\u00a0:\u00a0=\u00a0\u2211u\u2208Dw\u00a0(u)\u00a0+\u00a0\u2211v\u2208V\\D\u00a0min\u00a0{\u2211u\u2208Sw\u00a0(uv)\u00a0|S\u00a0\u2286\u00a0N\u00a0(v)\u00a0\u2229\u00a0D, |S|\u00a0=\u00a0k}. First, we put forward an integer linear programming model with a polynomial number of constrains for the WKD problem. If G is a cycle, we design a dynamic algorithm to determine its exact weighted 2-domination number. If G is a tree, we give a label algorithm to determine its exact weighted 2-domination number. Combining a primal-dual method and a greedy method, we put forward an approximation algorithm for general fuzzy network on the WKD problem. Finally, we describe an application of the WKD problem to police camp problem.<\/jats:p>","DOI":"10.3233\/jifs-213120","type":"journal-article","created":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T11:32:09Z","timestamp":1676979129000},"page":"7643-7651","source":"Crossref","is-referenced-by-count":0,"title":["Weighted k-domination problem in fuzzy networks"],"prefix":"10.1177","volume":"44","author":[{"given":"Xue-gang","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, North China Electric Power University, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Moo Young","family":"Sohn","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Changwon National University, Changwon, South Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"De-xiang","family":"Ma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, North China Electric Power University, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","reference":[{"key":"10.3233\/JIFS-213120_ref2","first-page":"15","article-title":"Graffiti.pcon the 2-domination number of a graph","volume":"203","author":"DeLaVi\u00f1a","year":"2010","journal-title":"Congr. 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