<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>1405-5546</journal-id>
<journal-title><![CDATA[Computación y Sistemas]]></journal-title>
<abbrev-journal-title><![CDATA[Comp. y Sist.]]></abbrev-journal-title>
<issn>1405-5546</issn>
<publisher>
<publisher-name><![CDATA[Instituto Politécnico Nacional, Centro de Investigación en Computación]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S1405-55462023000401089</article-id>
<article-id pub-id-type="doi">10.13053/cys-27-4-4782</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Applying Genetic Algorithms to Validate a Conjecture in Graph Theory: The Minimum Dominating Set Problem]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Cervantes-Ojeda]]></surname>
<given-names><![CDATA[Jorge]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Gómez-Fuentes]]></surname>
<given-names><![CDATA[María C.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Fresán-Figueroa]]></surname>
<given-names><![CDATA[Julián A.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad Autónoma Metropolitana  ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Mexico</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>12</month>
<year>2023</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>12</month>
<year>2023</year>
</pub-date>
<volume>27</volume>
<numero>4</numero>
<fpage>1089</fpage>
<lpage>1097</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_arttext&amp;pid=S1405-55462023000401089&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_abstract&amp;pid=S1405-55462023000401089&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_pdf&amp;pid=S1405-55462023000401089&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract: This paper presents a case study where the interdisciplinary approach between artificial intelligence, specifically genetic algorithms, and discrete mathematics has been instrumental in verifying a conjecture in graph theory. The focus of our research lies in the Minimum Dominating Set (MDS) problem, which involves identifying a dominating set with the minimum cardinality for a given graph. While previous works have primarily aimed at utilizing genetic algorithms to efficiently find satisfactory solutions to the MDS problem, our study aims to discover the optimal solution. The Rank GA algorithm employed in our work possesses the ability to escape local optima while simultaneously conducting local search to refine the best available solution. Since the MDS problem is known to be NP-hard, the characteristics of Rank GA, coupled with the identification of regularities in one of the solutions, enabled us to verify the conjecture for graphs comprising 30, 80, 312, and 800 vertices. This research serves as a testament to the versatility of genetic algorithms, showcasing their utility not only in practical problem-solving but also in tackling theoretical challenges.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[GA for theoretical problems]]></kwd>
<kwd lng="en"><![CDATA[rank GA]]></kwd>
<kwd lng="en"><![CDATA[genetic algorithms]]></kwd>
<kwd lng="en"><![CDATA[minimum dominating set]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<nlm-citation citation-type="confpro">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cervantes]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Stephens]]></surname>
<given-names><![CDATA[C. R.]]></given-names>
</name>
</person-group>
<source><![CDATA[Rank based variation operators for genetic algorithms]]></source>
<year>2008</year>
<conf-name><![CDATA[ 10th annual conference on Genetic and evolutionary computation]]></conf-name>
<conf-loc> </conf-loc>
<page-range>905-12</page-range></nlm-citation>
</ref>
<ref id="B2">
<nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Hedar]]></surname>
<given-names><![CDATA[A. R.]]></given-names>
</name>
<name>
<surname><![CDATA[Ismail]]></surname>
<given-names><![CDATA[R.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Hybrid genetic algorithm for minimum dominating set problem]]></article-title>
<person-group person-group-type="editor">
<name>
<surname><![CDATA[Taniar]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
<name>
<surname><![CDATA[Gervasi]]></surname>
<given-names><![CDATA[O.]]></given-names>
</name>
<name>
<surname><![CDATA[Murgante]]></surname>
<given-names><![CDATA[B.]]></given-names>
</name>
<name>
<surname><![CDATA[Pardede]]></surname>
<given-names><![CDATA[E.]]></given-names>
</name>
<name>
<surname><![CDATA[Apduhan]]></surname>
<given-names><![CDATA[B.O.]]></given-names>
</name>
</person-group>
<source><![CDATA[Computational Science and Its Applications &#8211; ICCSA 2010, Lecture Notes in Computer Science]]></source>
<year>2010</year>
<volume>6019</volume>
<page-range>457-67</page-range></nlm-citation>
</ref>
<ref id="B3">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Couturier]]></surname>
<given-names><![CDATA[J. F.]]></given-names>
</name>
<name>
<surname><![CDATA[Heggernes]]></surname>
<given-names><![CDATA[P.]]></given-names>
</name>
<name>
<surname><![CDATA[Van&#8217;t-Hof]]></surname>
<given-names><![CDATA[P.]]></given-names>
</name>
<name>
<surname><![CDATA[Kratsch]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Minimal dominating sets in graph classes: combinatorial bounds and enumeration]]></article-title>
<source><![CDATA[Theoretical Computer Science]]></source>
<year>2013</year>
<volume>487</volume>
<page-range>82-94</page-range></nlm-citation>
</ref>
<ref id="B4">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Liu]]></surname>
<given-names><![CDATA[Z.]]></given-names>
</name>
<name>
<surname><![CDATA[Wang]]></surname>
<given-names><![CDATA[B.]]></given-names>
</name>
<name>
<surname><![CDATA[Guo]]></surname>
<given-names><![CDATA[L.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A survey on connected dominating set construction algorithm for wireless sensor networks]]></article-title>
<source><![CDATA[Information Technology Journal]]></source>
<year>2010</year>
<volume>9</volume>
<numero>6</numero>
<issue>6</issue>
<page-range>1081-92</page-range></nlm-citation>
</ref>
<ref id="B5">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bai]]></surname>
<given-names><![CDATA[X.]]></given-names>
</name>
<name>
<surname><![CDATA[Zhao]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
<name>
<surname><![CDATA[Bai]]></surname>
<given-names><![CDATA[S.]]></given-names>
</name>
<name>
<surname><![CDATA[Wang]]></surname>
<given-names><![CDATA[Q.]]></given-names>
</name>
<name>
<surname><![CDATA[Li]]></surname>
<given-names><![CDATA[W.]]></given-names>
</name>
<name>
<surname><![CDATA[Mu]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Minimum connected dominating sets in heterogeneous 3D wireless ad hoc networks]]></article-title>
<source><![CDATA[Ad Hoc Networks]]></source>
<year>2020</year>
<volume>97</volume>
</nlm-citation>
</ref>
<ref id="B6">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bouamama]]></surname>
<given-names><![CDATA[S.]]></given-names>
</name>
<name>
<surname><![CDATA[Blum]]></surname>
<given-names><![CDATA[C.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[An improved greedy heuristic for the minimum positive influence dominating set problem in social networks]]></article-title>
<source><![CDATA[Algorithms]]></source>
<year>2021</year>
<volume>14</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>79</page-range></nlm-citation>
</ref>
<ref id="B7">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Chalupa]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[An order-based algorithm for minimum dominating set with application in graph mining]]></article-title>
<source><![CDATA[Information Sciences]]></source>
<year>2018</year>
<volume>426</volume>
<page-range>101-16</page-range></nlm-citation>
</ref>
<ref id="B8">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Dinh]]></surname>
<given-names><![CDATA[T. N.]]></given-names>
</name>
<name>
<surname><![CDATA[Shen]]></surname>
<given-names><![CDATA[Y.]]></given-names>
</name>
<name>
<surname><![CDATA[Nguyen]]></surname>
<given-names><![CDATA[D. T.]]></given-names>
</name>
<name>
<surname><![CDATA[Thai]]></surname>
<given-names><![CDATA[M. T.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[On the approximability of positive influence dominating set in social networks]]></article-title>
<source><![CDATA[Journal of Combinatorial Optimization]]></source>
<year>2014</year>
<volume>27</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>487-503</page-range></nlm-citation>
</ref>
<ref id="B9">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Levin]]></surname>
<given-names><![CDATA[M. S.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Note on dominating set problems]]></article-title>
<source><![CDATA[Journal of Communications Technology and Electronics]]></source>
<year>2021</year>
<volume>66</volume>
<numero>^s1</numero>
<issue>^s1</issue>
<supplement>1</supplement>
<page-range>S8-S22</page-range></nlm-citation>
</ref>
<ref id="B10">
<nlm-citation citation-type="confpro">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Li]]></surname>
<given-names><![CDATA[B.]]></given-names>
</name>
<name>
<surname><![CDATA[Zhang]]></surname>
<given-names><![CDATA[X.]]></given-names>
</name>
<name>
<surname><![CDATA[Cai]]></surname>
<given-names><![CDATA[S.]]></given-names>
</name>
<name>
<surname><![CDATA[Lin]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Wang]]></surname>
<given-names><![CDATA[Y.]]></given-names>
</name>
<name>
<surname><![CDATA[Blum]]></surname>
<given-names><![CDATA[C.]]></given-names>
</name>
</person-group>
<source><![CDATA[NuCDS: An efficient local search algorithm for minimum connected dominating set]]></source>
<year>2021</year>
<conf-name><![CDATA[ Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence]]></conf-name>
<conf-loc> </conf-loc>
<page-range>1503-10</page-range></nlm-citation>
</ref>
<ref id="B11">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Pinacho-Davidson]]></surname>
<given-names><![CDATA[P.]]></given-names>
</name>
<name>
<surname><![CDATA[Blum]]></surname>
<given-names><![CDATA[C.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Barrakuda: A hybrid evolutionary algorithm for minimum capacitated dominating set problem]]></article-title>
<source><![CDATA[Mathematics]]></source>
<year>2020</year>
<volume>8</volume>
<numero>11</numero>
<issue>11</issue>
<page-range>1858</page-range></nlm-citation>
</ref>
<ref id="B12">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Fomin]]></surname>
<given-names><![CDATA[F. V.]]></given-names>
</name>
<name>
<surname><![CDATA[Grandoni]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
<name>
<surname><![CDATA[Pyatkin]]></surname>
<given-names><![CDATA[A. V.]]></given-names>
</name>
<name>
<surname><![CDATA[Stepanov]]></surname>
<given-names><![CDATA[A. A.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications]]></article-title>
<source><![CDATA[ACM Transactions on Algorithms (TALG)]]></source>
<year>2008</year>
<volume>5</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>1-17</page-range></nlm-citation>
</ref>
<ref id="B13">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Mitchell]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<source><![CDATA[Introduction to genetic algorithms]]></source>
<year>1996</year>
<publisher-name><![CDATA[MIT Press]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B14">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[De Jong]]></surname>
<given-names><![CDATA[K. A.]]></given-names>
</name>
<name>
<surname><![CDATA[Spears]]></surname>
<given-names><![CDATA[W. M.]]></given-names>
</name>
</person-group>
<source><![CDATA[Using genetic algorithms to solve NP-complete problems]]></source>
<year>1989</year>
<page-range>124-32</page-range><publisher-name><![CDATA[ICGA]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B15">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Jakobs]]></surname>
<given-names><![CDATA[S.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[On genetic algorithms for the packing of polygons]]></article-title>
<source><![CDATA[European journal of operational research]]></source>
<year>1996</year>
<volume>88</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>165-81</page-range></nlm-citation>
</ref>
<ref id="B16">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Pourrajabian]]></surname>
<given-names><![CDATA[A.]]></given-names>
</name>
<name>
<surname><![CDATA[Ebrahimi]]></surname>
<given-names><![CDATA[R.]]></given-names>
</name>
<name>
<surname><![CDATA[Mirzaei]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
<name>
<surname><![CDATA[Shams]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Applying genetic algorithms for solving nonlinear algebraic equations]]></article-title>
<source><![CDATA[Applied Mathematics and Computation]]></source>
<year>2013</year>
<volume>219</volume>
<numero>24</numero>
<issue>24</issue>
<page-range>11483-94</page-range></nlm-citation>
</ref>
<ref id="B17">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cervantes-Ojeda]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Gómez-Fuentes]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
<name>
<surname><![CDATA[González-Moreno]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
<name>
<surname><![CDATA[Olsen]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Rainbow connectivity using a rank genetic algorithm: Moore cages with girth six]]></article-title>
<source><![CDATA[Journal of Applied Mathematics]]></source>
<year>2019</year>
<volume>2019</volume>
</nlm-citation>
</ref>
<ref id="B18">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[García-Altamirano]]></surname>
<given-names><![CDATA[J. C.]]></given-names>
</name>
<name>
<surname><![CDATA[Olsen]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
<name>
<surname><![CDATA[Cervantes-Ojeda]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[How to construct the symmetric cycle of length 5 using Hajós construction with an adapted rank genetic algorithm]]></article-title>
<source><![CDATA[Discrete Mathematics &amp; Theoretical Computer Science]]></source>
<year>2023</year>
<volume>25</volume>
<numero>1</numero>
<issue>1</issue>
</nlm-citation>
</ref>
<ref id="B19">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cervantes]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Stephens]]></surname>
<given-names><![CDATA[C. R.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Limitations of existing mutation rate heuristics and how a rank GA overcomes them]]></article-title>
<source><![CDATA[IEEE Transactions on Evolutionary Computation]]></source>
<year>2008</year>
<volume>13</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>369-97</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
