{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:43:24Z","timestamp":1747190604168,"version":"3.40.5"},"reference-count":12,"publisher":"Wiley","license":[{"start":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T00:00:00Z","timestamp":1611532800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Tishreen University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2021,1,25]]},"abstract":"<jats:p>An eternal dominating set of a graph <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                        <a:mi>G<\/a:mi>\n                     <\/a:math>\n                  <\/jats:inline-formula> is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks; an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the \u201call guards move\u201d of the eternal dominating set problem, in which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The \u201call guards move model\u201d is called the <jats:inline-formula>\n                     <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                        <c:mi>m<\/c:mi>\n                     <\/c:math>\n                  <\/jats:inline-formula>-eternal domination model. The size of the smallest <jats:inline-formula>\n                     <e:math xmlns:e=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <e:mi>m<\/e:mi>\n                     <\/e:math>\n                  <\/jats:inline-formula>-eternal dominating set is called the <jats:inline-formula>\n                     <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <g:mi>m<\/g:mi>\n                     <\/g:math>\n                  <\/jats:inline-formula>-eternal domination number and is denoted by <jats:inline-formula>\n                     <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <i:msubsup>\n                           <i:mrow>\n                              <i:mi>\u03b3<\/i:mi>\n                           <\/i:mrow>\n                           <i:mrow>\n                              <i:mi>m<\/i:mi>\n                           <\/i:mrow>\n                           <i:mrow>\n                              <i:mo>\u221e<\/i:mo>\n                           <\/i:mrow>\n                        <\/i:msubsup>\n                        <i:mfenced open=\"(\" close=\")\">\n                           <i:mrow>\n                              <i:mi>G<\/i:mi>\n                           <\/i:mrow>\n                        <\/i:mfenced>\n                     <\/i:math>\n                  <\/jats:inline-formula>. In this paper, we find <jats:inline-formula>\n                     <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <m:msubsup>\n                           <m:mrow>\n                              <m:mi>\u03b3<\/m:mi>\n                           <\/m:mrow>\n                           <m:mrow>\n                              <m:mi>m<\/m:mi>\n                           <\/m:mrow>\n                           <m:mrow>\n                              <m:mo>\u221e<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:msubsup>\n                        <m:mfenced open=\"(\" close=\")\">\n                           <m:mrow>\n                              <m:mi>P<\/m:mi>\n                              <m:mfenced open=\"(\" close=\")\">\n                                 <m:mrow>\n                                    <m:mi>n<\/m:mi>\n                                    <m:mo>,<\/m:mo>\n                                    <m:mn>1<\/m:mn>\n                                 <\/m:mrow>\n                              <\/m:mfenced>\n                           <\/m:mrow>\n                        <\/m:mfenced>\n                     <\/m:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <s:math xmlns:s=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                        <s:msubsup>\n                           <s:mrow>\n                              <s:mi>\u03b3<\/s:mi>\n                           <\/s:mrow>\n                           <s:mrow>\n                              <s:mi>m<\/s:mi>\n                           <\/s:mrow>\n                           <s:mrow>\n                              <s:mo>\u221e<\/s:mo>\n                           <\/s:mrow>\n                        <\/s:msubsup>\n                        <s:mfenced open=\"(\" close=\")\">\n                           <s:mrow>\n                              <s:mi>P<\/s:mi>\n                              <s:mfenced open=\"(\" close=\")\">\n                                 <s:mrow>\n                                    <s:mi>n<\/s:mi>\n                                    <s:mo>,<\/s:mo>\n                                    <s:mn>3<\/s:mn>\n                                 <\/s:mrow>\n                              <\/s:mfenced>\n                           <\/s:mrow>\n                        <\/s:mfenced>\n                     <\/s:math>\n                  <\/jats:inline-formula> for <jats:inline-formula>\n                     <y:math xmlns:y=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\">\n                        <y:mi>n<\/y:mi>\n                        <y:mo>\u2261<\/y:mo>\n                        <y:mn>0<\/y:mn>\n                        <y:mtext>\u2009<\/y:mtext>\n                        <y:mfenced open=\"(\" close=\")\">\n                           <y:mrow>\n                              <y:mi mathvariant=\"normal\">mod<\/y:mi>\n                              <y:mtext>\u2009<\/y:mtext>\n                              <y:mn>4<\/y:mn>\n                           <\/y:mrow>\n                        <\/y:mfenced>\n                     <\/y:math>\n                  <\/jats:inline-formula>. We also find upper bounds for <jats:inline-formula>\n                     <db:math xmlns:db=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\">\n                        <db:msubsup>\n                           <db:mrow>\n                              <db:mi>\u03b3<\/db:mi>\n                           <\/db:mrow>\n                           <db:mrow>\n                              <db:mi>m<\/db:mi>\n                           <\/db:mrow>\n                           <db:mrow>\n                              <db:mo>\u221e<\/db:mo>\n                           <\/db:mrow>\n                        <\/db:msubsup>\n                        <db:mfenced open=\"(\" close=\")\">\n                           <db:mrow>\n                              <db:mi>P<\/db:mi>\n                              <db:mfenced open=\"(\" close=\")\">\n                                 <db:mrow>\n                                    <db:mi>n<\/db:mi>\n                                    <db:mo>,<\/db:mo>\n                                    <db:mn>2<\/db:mn>\n                                 <\/db:mrow>\n                              <\/db:mfenced>\n                           <\/db:mrow>\n                        <\/db:mfenced>\n                     <\/db:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <jb:math xmlns:jb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\">\n                        <jb:msubsup>\n                           <jb:mrow>\n                              <jb:mi>\u03b3<\/jb:mi>\n                           <\/jb:mrow>\n                           <jb:mrow>\n                              <jb:mi>m<\/jb:mi>\n                           <\/jb:mrow>\n                           <jb:mrow>\n                              <jb:mo>\u221e<\/jb:mo>\n                           <\/jb:mrow>\n                        <\/jb:msubsup>\n                        <jb:mfenced open=\"(\" close=\")\">\n                           <jb:mrow>\n                              <jb:mi>P<\/jb:mi>\n                              <jb:mfenced open=\"(\" close=\")\">\n                                 <jb:mrow>\n                                    <jb:mi>n<\/jb:mi>\n                                    <jb:mo>,<\/jb:mo>\n                                    <jb:mn>3<\/jb:mn>\n                                 <\/jb:mrow>\n                              <\/jb:mfenced>\n                           <\/jb:mrow>\n                        <\/jb:mfenced>\n                     <\/jb:math>\n                  <\/jats:inline-formula> when <jats:inline-formula>\n                     <pb:math xmlns:pb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\">\n                        <pb:mi>n<\/pb:mi>\n                     <\/pb:math>\n                  <\/jats:inline-formula> is arbitrary.<\/jats:p>","DOI":"10.1155\/2021\/6627272","type":"journal-article","created":{"date-parts":[[2021,1,28]],"date-time":"2021-01-28T01:35:16Z","timestamp":1611797716000},"page":"1-10","source":"Crossref","is-referenced-by-count":0,"title":["Eternal Domination of Generalized Petersen Graph"],"prefix":"10.1155","volume":"2021","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8515-6498","authenticated-orcid":true,"given":"Ramy","family":"Shaheen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Tishreen University, Latakia, Syria"}]},{"given":"Ali","family":"Kassem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Tishreen University, Latakia, Syria"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.2298\/AADM151109021K"},{"key":"2","first-page":"179","article-title":"Infinite order domination in graphs","volume":"50","author":"A. P. Burger","year":"2004","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"3","first-page":"169","article-title":"Eternal security in graphs","volume":"52","author":"W. Goddard","year":"2005","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"4","first-page":"156","article-title":"Eternal domination on 3\u00d7 n grid graphs","volume":"61","author":"S. Finbow","year":"2015","journal-title":"The Australasian Journal of Combinatorics"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.11575\/cdm.v12i1.62531"},{"issue":"2","key":"6","first-page":"123","article-title":"Chemical indices of generalized Petersen graph","volume":"47","author":"Y. Liu","year":"2017","journal-title":"International Journal of Applied Mathematics"},{"key":"7","first-page":"47","article-title":"Eternal protection in grid graphs","volume":"91","author":"J. Goldwasser","year":"2013","journal-title":"Utilitas Mathematica"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.12988\/ijcms.2007.07122"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.4236\/ojdm.2019.93008"},{"key":"10","article-title":"Domination and eternal domination of generalized Jahangir graph","author":"R. Shaheen","year":"2019","journal-title":"Journal of applied Mathematics"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2009.01.018"},{"key":"12","first-page":"239","article-title":"Independence number of generalized Petersen graphs","volume":"124","author":"N. Besharati","year":"2016","journal-title":"Ars Combinatoria"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2021\/6627272.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2021\/6627272.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2021\/6627272.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,28]],"date-time":"2021-01-28T01:35:23Z","timestamp":1611797723000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/jam\/2021\/6627272\/"}},"subtitle":[],"editor":[{"given":"Jian G.","family":"Zhou","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2021,1,25]]},"references-count":12,"alternative-id":["6627272","6627272"],"URL":"https:\/\/doi.org\/10.1155\/2021\/6627272","relation":{},"ISSN":["1687-0042","1110-757X"],"issn-type":[{"type":"electronic","value":"1687-0042"},{"type":"print","value":"1110-757X"}],"subject":[],"published":{"date-parts":[[2021,1,25]]}}}