Guillermo De Ita Luna, Cristina López-Ramírez, Ana E. De Ita-Varela, Jorge E. Gutiérrez-Gómez
{"title":"平面图形着色的一种启发式算法","authors":"Guillermo De Ita Luna, Cristina López-Ramírez, Ana E. De Ita-Varela, Jorge E. Gutiérrez-Gómez","doi":"10.1016/j.entcs.2020.10.008","DOIUrl":null,"url":null,"abstract":"<div><p>We present an algorithm for the coloring of planar graphs based on the construction of a maximal independent set <em>S</em> of the input graph. The maximal independent set <em>S</em> must fulfill certain characteristics. For example, <em>S</em> contains the vertex that appears in a maximum number of odd cycles of <em>G</em>. The construction of <em>S</em> considers the internal-face graph of the input graph <em>G</em> in order to select each vertex belonging to a maximal number of odd faces of <em>G</em>.</p><p>The traversing in pre-order on the internal-face graph <em>G</em><sub><em>f</em></sub> of the input planar graph <em>G</em> provides us of a strategy for the construction of partial maximal independent sets of critical regions of <em>G</em><sub><em>f</em></sub>. Thus, the union of these partial maximal independent sets forms a maximal independent set <em>S</em> of <em>G</em>. This allows us to color first the vertices that are crucial for decomposing <em>G</em> in a graph (<em>G − S</em>), which is a polygonal tree, and therefore, is 3-colorable.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":"354 ","pages":"Pages 91-105"},"PeriodicalIF":0.0000,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2020.10.008","citationCount":"0","resultStr":"{\"title\":\"A Heuristic for the Coloring of Planar Graphs\",\"authors\":\"Guillermo De Ita Luna, Cristina López-Ramírez, Ana E. De Ita-Varela, Jorge E. Gutiérrez-Gómez\",\"doi\":\"10.1016/j.entcs.2020.10.008\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We present an algorithm for the coloring of planar graphs based on the construction of a maximal independent set <em>S</em> of the input graph. The maximal independent set <em>S</em> must fulfill certain characteristics. For example, <em>S</em> contains the vertex that appears in a maximum number of odd cycles of <em>G</em>. The construction of <em>S</em> considers the internal-face graph of the input graph <em>G</em> in order to select each vertex belonging to a maximal number of odd faces of <em>G</em>.</p><p>The traversing in pre-order on the internal-face graph <em>G</em><sub><em>f</em></sub> of the input planar graph <em>G</em> provides us of a strategy for the construction of partial maximal independent sets of critical regions of <em>G</em><sub><em>f</em></sub>. Thus, the union of these partial maximal independent sets forms a maximal independent set <em>S</em> of <em>G</em>. This allows us to color first the vertices that are crucial for decomposing <em>G</em> in a graph (<em>G − S</em>), which is a polygonal tree, and therefore, is 3-colorable.</p></div>\",\"PeriodicalId\":38770,\"journal\":{\"name\":\"Electronic Notes in Theoretical Computer Science\",\"volume\":\"354 \",\"pages\":\"Pages 91-105\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.entcs.2020.10.008\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Notes in Theoretical Computer Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1571066120300840\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Computer Science\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Notes in Theoretical Computer Science","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1571066120300840","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Computer Science","Score":null,"Total":0}
We present an algorithm for the coloring of planar graphs based on the construction of a maximal independent set S of the input graph. The maximal independent set S must fulfill certain characteristics. For example, S contains the vertex that appears in a maximum number of odd cycles of G. The construction of S considers the internal-face graph of the input graph G in order to select each vertex belonging to a maximal number of odd faces of G.
The traversing in pre-order on the internal-face graph Gf of the input planar graph G provides us of a strategy for the construction of partial maximal independent sets of critical regions of Gf. Thus, the union of these partial maximal independent sets forms a maximal independent set S of G. This allows us to color first the vertices that are crucial for decomposing G in a graph (G − S), which is a polygonal tree, and therefore, is 3-colorable.
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