{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T06:42:59Z","timestamp":1648881779108},"reference-count":10,"publisher":"EDP Sciences","issue":"1","license":[{"start":{"date-parts":[[2019,2,15]],"date-time":"2019-02-15T00:00:00Z","timestamp":1550188800000},"content-version":"vor","delay-in-days":45,"URL":"https:\/\/www.edpsciences.org\/en\/authors\/copyright-and-licensing"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2018,5,21]]},"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>The interconnection networks are modeled by means of graphs to determine the reliability and vulnerability. There are lots of parameters that are used to determine vulnerability. The average covering number is one of them which is denoted by $ \\overline{\\beta }(G)$, where <jats:italic>G<\/jats:italic> is simple, connected and undirected graph of order <jats:italic>n<\/jats:italic>\u00a0\u2265\u00a02. In a graph <jats:italic>G<\/jats:italic>\u00a0=\u00a0(<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>), <jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>)) a subset \n\t\t\n\t\t\n\t\t$ {S}_v\\subseteq V(G)$\n\n\n\t\tof vertices is called a cover set of <jats:italic>G<\/jats:italic> with respect to <jats:italic>v<\/jats:italic> or a local covering set of vertex <jats:italic>v<\/jats:italic>, if each edge of the graph is incident to at least one vertex of <jats:italic>S<\/jats:italic><jats:sub><jats:italic>v<\/jats:italic><\/jats:sub>. The local covering number with respect to <jats:italic>v<\/jats:italic> is the minimum cardinality of among the <jats:italic>S<\/jats:italic><jats:sub><jats:italic>v<\/jats:italic><\/jats:sub> sets and denoted by <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>v<\/jats:italic><\/jats:sub>. The average covering number of a graph <jats:italic>G<\/jats:italic> is defined as\n\t\t\n\t\t<jats:disp-formula id=\"FD1\">\n\t\t<jats:italic>\u03b2<\/jats:italic>\u0305(<jats:italic>G<\/jats:italic>) = 1\/|<jats:italic>v<\/jats:italic>(<jats:italic>G<\/jats:italic>)| \u2211<jats:sub><jats:italic>\u03bd<\/jats:italic>\u2208<jats:italic>v<\/jats:italic>(<jats:italic>G<\/jats:italic>)<\/jats:sub><jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>\u03bd<\/jats:italic><\/jats:sub>\n\t\t<\/jats:disp-formula>\n\t\t\n\t\t\t\t\t\t\n\t\t\n      In this paper, the average covering numbers of <jats:italic>k<\/jats:italic>th power of a cycle \n\t  $ {C}_n^k$ and <jats:italic>P<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> \u25a1 <jats:italic>P<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub>, <jats:italic>P<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> \u25a1 <jats:italic>C<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub>, cartesian product of <jats:italic>P<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> and <jats:italic>P<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub>, cartesian product of <jats:italic>P<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> and <jats:italic>C<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub> are given, respectively.<\/jats:p>","DOI":"10.1051\/ro\/2018044","type":"journal-article","created":{"date-parts":[[2018,5,31]],"date-time":"2018-05-31T08:23:13Z","timestamp":1527754993000},"page":"261-268","source":"Crossref","is-referenced-by-count":0,"title":["Average covering number for some graphs"],"prefix":"10.1051","volume":"53","author":[{"given":"D.","family":"Do\u011fan Durgun","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ali","family":"Bagatarhan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2019,2,15]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/S0012-365X(01)00180-7","volume":"252","author":"Beineke","year":"2002","journal-title":"Discrete Math."},{"key":"R2","doi-asserted-by":"crossref","unstructured":"Chartrand G., Lesniak L. and Zhang P., Textbooks in Mathematics, Graphs & Digraphs. 6th edition. 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