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Appl. Math."],"published-print":{"date-parts":[[2025,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>A function <jats:italic>f<\/jats:italic> that assigns values from the set <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\{0, 1, 2\\}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> to each vertex of a graph <jats:italic>G<\/jats:italic> is called a 2-rainbow independent dominating function, if the vertices assigned the value 1 form an independent set, the vertices assigned the value 2 form another independent set, and every vertex to which 0 is assigned has at least one neighbor in each of the mentioned independent sets. The weight of this function is the total number of vertices assigned nonzero values. The 2-rainbow independent domination number of <jats:italic>G<\/jats:italic>, <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\gamma _{\\textrm{ri}2}(G)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>\u03b3<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mtext>ri<\/mml:mtext>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>G<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, is the minimum weight of such a function. Motivated by a real-life application, we study the 2-rainbow independent domination number of the complementary prism <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$G \\overline{G}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>G<\/mml:mi>\n                    <mml:mover>\n                      <mml:mi>G<\/mml:mi>\n                      <mml:mo>\u00af<\/mml:mo>\n                    <\/mml:mover>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> of a graph <jats:italic>G<\/jats:italic>, which is constructed by taking <jats:italic>G<\/jats:italic> and its complement <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\overline{G}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mover>\n                    <mml:mi>G<\/mml:mi>\n                    <mml:mo>\u00af<\/mml:mo>\n                  <\/mml:mover>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, and then adding edges between corresponding vertices. We provide tight bounds for <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\gamma _{\\textrm{ri}2}(G\\overline{G})$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>\u03b3<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mtext>ri<\/mml:mtext>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>G<\/mml:mi>\n                      <mml:mover>\n                        <mml:mi>G<\/mml:mi>\n                        <mml:mo>\u00af<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, and characterize graphs for which the lower bound, i.e. <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\max \\{\\gamma _{\\textrm{ri}2}(G), \\gamma _{\\textrm{ri}2}(\\overline{G})\\}+1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>max<\/mml:mo>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>\u03b3<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mtext>ri<\/mml:mtext>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>G<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>\u03b3<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mtext>ri<\/mml:mtext>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mover>\n                        <mml:mi>G<\/mml:mi>\n                        <mml:mo>\u00af<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>}<\/mml:mo>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, is attained. The obtained results can, in practice, enable the prediction of the cost estimate for a given communication or surveillance network.<\/jats:p>","DOI":"10.1007\/s40314-025-03189-9","type":"journal-article","created":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T15:04:01Z","timestamp":1744383841000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["2-rainbow independent domination in complementary prisms"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-0989-2450","authenticated-orcid":false,"given":"Dragana","family":"Bo\u017eovi\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gordana","family":"Radi\u0107","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aleksandra","family":"Tepeh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,11]]},"reference":[{"key":"3189_CR1","doi-asserted-by":"publisher","first-page":"214","DOI":"10.1016\/j.amc.2018.12.009","volume":"349","author":"S Brezovnik","year":"2019","unstructured":"Brezovnik S, \u0160umenjak T Kraner (2019) Complexity of k -rainbow independent domination and some results on the lexicographic product of graphs. 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