{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:45:54Z","timestamp":1747190754722,"version":"3.40.5"},"reference-count":7,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,8,11]],"date-time":"2022-08-11T00:00:00Z","timestamp":1660176000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100019812","name":"Tishreen University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100019812","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2022,8,11]]},"abstract":"<jats:p>An irreversible conversion process is a dynamic process on a graph where a one-way change of state (from state 0 to state 1) is applied on the vertices if they satisfy a conversion rule that is determined at the beginning of the study. The irreversible <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                        <a:mi>k<\/a:mi>\n                     <\/a:math>\n                  <\/jats:inline-formula>-threshold conversion process on a graph <jats:inline-formula>\n                     <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <c:mi>G<\/c:mi>\n                        <c:mo>=<\/c:mo>\n                        <c:mfenced open=\"(\" close=\")\">\n                           <c:mrow>\n                              <c:mi>V<\/c:mi>\n                              <c:mo>,<\/c:mo>\n                              <c:mi>E<\/c:mi>\n                           <\/c:mrow>\n                        <\/c:mfenced>\n                     <\/c:math>\n                  <\/jats:inline-formula> is an iterative process which begins by choosing a set <jats:inline-formula>\n                     <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <g:msub>\n                           <g:mrow>\n                              <g:mi>S<\/g:mi>\n                           <\/g:mrow>\n                           <g:mrow>\n                              <g:mn>0<\/g:mn>\n                           <\/g:mrow>\n                        <\/g:msub>\n                        <g:mo>\u2286<\/g:mo>\n                        <g:mi>V<\/g:mi>\n                     <\/g:math>\n                  <\/jats:inline-formula>, and for each step <jats:inline-formula>\n                     <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <i:mi>t<\/i:mi>\n                        <i:mfenced open=\"(\" close=\")\">\n                           <i:mrow>\n                              <i:mi>t<\/i:mi>\n                              <i:mo>=<\/i:mo>\n                              <i:mn>1<\/i:mn>\n                              <i:mo>,<\/i:mo>\n                              <i:mn>2<\/i:mn>\n                              <i:mo>,<\/i:mo>\n                              <i:mo>\u22ef<\/i:mo>\n                              <i:mo>,<\/i:mo>\n                           <\/i:mrow>\n                        <\/i:mfenced>\n                        <i:mo>,<\/i:mo>\n                     <\/i:math>\n                  <\/jats:inline-formula>\n                  <jats:inline-formula>\n                     <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <m:msub>\n                           <m:mrow>\n                              <m:mi>S<\/m:mi>\n                           <\/m:mrow>\n                           <m:mrow>\n                              <m:mi>t<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:msub>\n                     <\/m:math>\n                  <\/jats:inline-formula> is obtained from <jats:inline-formula>\n                     <o:math xmlns:o=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                        <o:msub>\n                           <o:mrow>\n                              <o:mi>S<\/o:mi>\n                           <\/o:mrow>\n                           <o:mrow>\n                              <o:mi>t<\/o:mi>\n                              <o:mo>\u2212<\/o:mo>\n                              <o:mn>1<\/o:mn>\n                           <\/o:mrow>\n                        <\/o:msub>\n                     <\/o:math>\n                  <\/jats:inline-formula> by adjoining all vertices that have at least <jats:inline-formula>\n                     <q:math xmlns:q=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\">\n                        <q:mi>k<\/q:mi>\n                     <\/q:math>\n                  <\/jats:inline-formula> neighbors in <jats:inline-formula>\n                     <s:math xmlns:s=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\">\n                        <s:msub>\n                           <s:mrow>\n                              <s:mi>S<\/s:mi>\n                           <\/s:mrow>\n                           <s:mrow>\n                              <s:mi>t<\/s:mi>\n                              <s:mo>\u2212<\/s:mo>\n                              <s:mn>1<\/s:mn>\n                           <\/s:mrow>\n                        <\/s:msub>\n                     <\/s:math>\n                  <\/jats:inline-formula>. <jats:inline-formula>\n                     <u:math xmlns:u=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\">\n                        <u:msub>\n                           <u:mrow>\n                              <u:mi>S<\/u:mi>\n                           <\/u:mrow>\n                           <u:mrow>\n                              <u:mn>0<\/u:mn>\n                           <\/u:mrow>\n                        <\/u:msub>\n                     <\/u:math>\n                  <\/jats:inline-formula> is called the seed set of the <jats:inline-formula>\n                     <w:math xmlns:w=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\">\n                        <w:mi>k<\/w:mi>\n                     <\/w:math>\n                  <\/jats:inline-formula>-threshold conversion process, and if <jats:inline-formula>\n                     <y:math xmlns:y=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\">\n                        <y:msub>\n                           <y:mrow>\n                              <y:mi>S<\/y:mi>\n                           <\/y:mrow>\n                           <y:mrow>\n                              <y:mi>t<\/y:mi>\n                           <\/y:mrow>\n                        <\/y:msub>\n                        <y:mo>=<\/y:mo>\n                        <y:mi>V<\/y:mi>\n                        <y:mfenced open=\"(\" close=\")\">\n                           <y:mrow>\n                              <y:mi>G<\/y:mi>\n                           <\/y:mrow>\n                        <\/y:mfenced>\n                     <\/y:math>\n                  <\/jats:inline-formula> for some <jats:inline-formula>\n                     <cb:math xmlns:cb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\">\n                        <cb:mi>t<\/cb:mi>\n                        <cb:mo>\u2265<\/cb:mo>\n                        <cb:mn>0<\/cb:mn>\n                     <\/cb:math>\n                  <\/jats:inline-formula>, then <jats:inline-formula>\n                     <eb:math xmlns:eb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\">\n                        <eb:msub>\n                           <eb:mrow>\n                              <eb:mi>S<\/eb:mi>\n                           <\/eb:mrow>\n                           <eb:mrow>\n                              <eb:mn>0<\/eb:mn>\n                           <\/eb:mrow>\n                        <\/eb:msub>\n                     <\/eb:math>\n                  <\/jats:inline-formula> is an irreversible <jats:inline-formula>\n                     <gb:math xmlns:gb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\">\n                        <gb:mi>k<\/gb:mi>\n                     <\/gb:math>\n                  <\/jats:inline-formula>-threshold conversion set (IkCS) of <jats:inline-formula>\n                     <ib:math xmlns:ib=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\">\n                        <ib:mi>G<\/ib:mi>\n                     <\/ib:math>\n                  <\/jats:inline-formula>. The <jats:inline-formula>\n                     <kb:math xmlns:kb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\">\n                        <kb:mi>k<\/kb:mi>\n                     <\/kb:math>\n                  <\/jats:inline-formula>-threshold conversion number of <jats:inline-formula>\n                     <mb:math xmlns:mb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\">\n                        <mb:mi>G<\/mb:mi>\n                     <\/mb:math>\n                  <\/jats:inline-formula> (denoted by (<jats:inline-formula>\n                     <ob:math xmlns:ob=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\">\n                        <ob:msub>\n                           <ob:mrow>\n                              <ob:mi>C<\/ob:mi>\n                           <\/ob:mrow>\n                           <ob:mrow>\n                              <ob:mi>k<\/ob:mi>\n                           <\/ob:mrow>\n                        <\/ob:msub>\n                        <ob:mfenced open=\"(\" close=\")\">\n                           <ob:mrow>\n                              <ob:mi>G<\/ob:mi>\n                           <\/ob:mrow>\n                        <\/ob:mfenced>\n                     <\/ob:math>\n                  <\/jats:inline-formula>) is the minimum cardinality of all the IkCSs of <jats:inline-formula>\n                     <sb:math xmlns:sb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M20\">\n                        <sb:mi>G<\/sb:mi>\n                        <sb:mo>.<\/sb:mo>\n                     <\/sb:math>\n                  <\/jats:inline-formula> In this paper, we determine <jats:inline-formula>\n                     <ub:math xmlns:ub=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M21\">\n                        <ub:msub>\n                           <ub:mrow>\n                              <ub:mi>C<\/ub:mi>\n                           <\/ub:mrow>\n                           <ub:mrow>\n                              <ub:mn>2<\/ub:mn>\n                           <\/ub:mrow>\n                        <\/ub:msub>\n                        <ub:mfenced open=\"(\" close=\")\">\n                           <ub:mrow>\n                              <ub:mi>G<\/ub:mi>\n                           <\/ub:mrow>\n                        <\/ub:mfenced>\n                     <\/ub:math>\n                  <\/jats:inline-formula> for the circulant graph <jats:inline-formula>\n                     <yb:math xmlns:yb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M22\">\n                        <yb:msub>\n                           <yb:mrow>\n                              <yb:mi>C<\/yb:mi>\n                           <\/yb:mrow>\n                           <yb:mrow>\n                              <yb:mi>n<\/yb:mi>\n                           <\/yb:mrow>\n                        <\/yb:msub>\n                        <yb:mfenced open=\"(\" close=\")\">\n                           <yb:mrow>\n                              <yb:mfenced open=\"{\" close=\"}\">\n                                 <yb:mrow>\n                                    <yb:mn>1<\/yb:mn>\n                                    <yb:mo>,<\/yb:mo>\n                                    <yb:mtext>r<\/yb:mtext>\n                                 <\/yb:mrow>\n                              <\/yb:mfenced>\n                           <\/yb:mrow>\n                        <\/yb:mfenced>\n                     <\/yb:math>\n                  <\/jats:inline-formula> when <jats:inline-formula>\n                     <ec:math xmlns:ec=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M23\">\n                        <ec:mi>r<\/ec:mi>\n                     <\/ec:math>\n                  <\/jats:inline-formula> is arbitrary; we also find <jats:inline-formula>\n                     <gc:math xmlns:gc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M24\">\n                        <gc:msub>\n                           <gc:mrow>\n                              <gc:mi>C<\/gc:mi>\n                           <\/gc:mrow>\n                           <gc:mrow>\n                              <gc:mn>3<\/gc:mn>\n                           <\/gc:mrow>\n                        <\/gc:msub>\n                        <gc:mfenced open=\"(\" close=\")\">\n                           <gc:mrow>\n                              <gc:msub>\n                                 <gc:mrow>\n                                    <gc:mi>C<\/gc:mi>\n                                 <\/gc:mrow>\n                                 <gc:mrow>\n                                    <gc:mi>n<\/gc:mi>\n                                 <\/gc:mrow>\n                              <\/gc:msub>\n                              <gc:mfenced open=\"(\" close=\")\">\n                                 <gc:mrow>\n                                    <gc:mfenced open=\"{\" close=\"}\">\n                                       <gc:mrow>\n                                          <gc:mn>1<\/gc:mn>\n                                          <gc:mo>,<\/gc:mo>\n                                          <gc:mtext>r<\/gc:mtext>\n                                       <\/gc:mrow>\n                                    <\/gc:mfenced>\n                                 <\/gc:mrow>\n                              <\/gc:mfenced>\n                           <\/gc:mrow>\n                        <\/gc:mfenced>\n                     <\/gc:math>\n                  <\/jats:inline-formula> when <jats:inline-formula>\n                     <oc:math xmlns:oc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M25\">\n                        <oc:mi>r<\/oc:mi>\n                        <oc:mo>=<\/oc:mo>\n                        <oc:mn>2<\/oc:mn>\n                        <oc:mo>,<\/oc:mo>\n                        <oc:mn>3<\/oc:mn>\n                     <\/oc:math>\n                  <\/jats:inline-formula>. We also introduce an upper bound for <jats:inline-formula>\n                     <qc:math xmlns:qc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M26\">\n                        <qc:msub>\n                           <qc:mrow>\n                              <qc:mi>C<\/qc:mi>\n                           <\/qc:mrow>\n                           <qc:mrow>\n                              <qc:mn>3<\/qc:mn>\n                           <\/qc:mrow>\n                        <\/qc:msub>\n                        <qc:mfenced open=\"(\" close=\")\">\n                           <qc:mrow>\n                              <qc:msub>\n                                 <qc:mrow>\n                                    <qc:mi>C<\/qc:mi>\n                                 <\/qc:mrow>\n                                 <qc:mrow>\n                                    <qc:mi>n<\/qc:mi>\n                                 <\/qc:mrow>\n                              <\/qc:msub>\n                              <qc:mfenced open=\"(\" close=\")\">\n                                 <qc:mrow>\n                                    <qc:mfenced open=\"{\" close=\"}\">\n                                       <qc:mrow>\n                                          <qc:mn>1<\/qc:mn>\n                                          <qc:mo>,<\/qc:mo>\n                                          <qc:mn>4<\/qc:mn>\n                                       <\/qc:mrow>\n                                    <\/qc:mfenced>\n                                 <\/qc:mrow>\n                              <\/qc:mfenced>\n                           <\/qc:mrow>\n                        <\/qc:mfenced>\n                     <\/qc:math>\n                  <\/jats:inline-formula>. Finally, we suggest an upper bound for <jats:inline-formula>\n                     <yc:math xmlns:yc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M27\">\n                        <yc:msub>\n                           <yc:mrow>\n                              <yc:mi>C<\/yc:mi>\n                           <\/yc:mrow>\n                           <yc:mrow>\n                              <yc:mn>3<\/yc:mn>\n                           <\/yc:mrow>\n                        <\/yc:msub>\n                        <yc:mfenced open=\"(\" close=\")\">\n                           <yc:mrow>\n                              <yc:msub>\n                                 <yc:mrow>\n                                    <yc:mi>C<\/yc:mi>\n                                 <\/yc:mrow>\n                                 <yc:mrow>\n                                    <yc:mi>n<\/yc:mi>\n                                 <\/yc:mrow>\n                              <\/yc:msub>\n                              <yc:mfenced open=\"(\" close=\")\">\n                                 <yc:mrow>\n                                    <yc:mfenced open=\"{\" close=\"}\">\n                                       <yc:mrow>\n                                          <yc:mn>1<\/yc:mn>\n                                          <yc:mo>,<\/yc:mo>\n                                          <yc:mtext>r<\/yc:mtext>\n                                       <\/yc:mrow>\n                                    <\/yc:mfenced>\n                                 <\/yc:mrow>\n                              <\/yc:mfenced>\n                           <\/yc:mrow>\n                        <\/yc:mfenced>\n                     <\/yc:math>\n                  <\/jats:inline-formula> if <jats:inline-formula>\n                     <gd:math xmlns:gd=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M28\">\n                        <gd:mi>n<\/gd:mi>\n                        <gd:mo>\u2265<\/gd:mo>\n                        <gd:mn>2<\/gd:mn>\n                        <gd:mfenced open=\"(\" close=\")\">\n                           <gd:mrow>\n                              <gd:mi>r<\/gd:mi>\n                              <gd:mo>+<\/gd:mo>\n                              <gd:mn>1<\/gd:mn>\n                           <\/gd:mrow>\n                        <\/gd:mfenced>\n                     <\/gd:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <kd:math xmlns:kd=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M29\">\n                        <kd:mi>n<\/kd:mi>\n                        <kd:mo>\u2261<\/kd:mo>\n                        <kd:mn>0<\/kd:mn>\n                        <kd:mfenced open=\"(\" close=\")\">\n                           <kd:mrow>\n                              <kd:mi>mod<\/kd:mi>\n                              <kd:mtext>\u2009<\/kd:mtext>\n                              <kd:mn>2<\/kd:mn>\n                              <kd:mfenced open=\"(\" close=\")\">\n                                 <kd:mrow>\n                                    <kd:mi>r<\/kd:mi>\n                                    <kd:mo>+<\/kd:mo>\n                                    <kd:mn>1<\/kd:mn>\n                                 <\/kd:mrow>\n                              <\/kd:mfenced>\n                           <\/kd:mrow>\n                        <\/kd:mfenced>\n                     <\/kd:math>\n                  <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1155\/2022\/1250951","type":"journal-article","created":{"date-parts":[[2022,8,11]],"date-time":"2022-08-11T21:35:08Z","timestamp":1660253708000},"page":"1-14","source":"Crossref","is-referenced-by-count":1,"title":["Irreversible <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                     <a:mi>k<\/a:mi>\n                  <\/a:math>-Threshold Conversion Number of Circulant Graphs"],"prefix":"10.1155","volume":"2022","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, Lattakia, Syria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9747-9668","authenticated-orcid":true,"given":"Suhail","family":"Mahfud","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1275-9191","authenticated-orcid":true,"given":"Ali","family":"Kassem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"crossref","DOI":"10.21236\/AD0705364","volume-title":"Graph Theory","author":"F. Harary","year":"1969"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2008.09.012"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2011.03.029"},{"key":"4","first-page":"156","article-title":"Irreversible k-threshold and majority conversion processes on complete multipartite graphs and graph products","volume":"61","author":"S. S. Adams","year":"2015","journal-title":"The Australasian Journal of Combinatorics"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.23638\/DMTCS-19-3-5"},{"issue":"3","key":"6","first-page":"288","article-title":"Subgraph-avoiding minimum decycling sets and k-conversion sets in graphs","volume":"74","author":"M. D. Frances","year":"2019","journal-title":"The Australasian Journal of Combinatorics"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(79)90011-6"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2022\/1250951.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2022\/1250951.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2022\/1250951.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,11]],"date-time":"2022-08-11T21:35:15Z","timestamp":1660253715000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/jam\/2022\/1250951\/"}},"subtitle":[],"editor":[{"given":"A. R.","family":"Ashrafi","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2022,8,11]]},"references-count":7,"alternative-id":["1250951","1250951"],"URL":"https:\/\/doi.org\/10.1155\/2022\/1250951","relation":{},"ISSN":["1687-0042","1110-757X"],"issn-type":[{"type":"electronic","value":"1687-0042"},{"type":"print","value":"1110-757X"}],"subject":[],"published":{"date-parts":[[2022,8,11]]}}}