{"title":"论图的路径数计算","authors":"F. Botler, R. Cano, M. Sambinelli","doi":"10.1016/j.entcs.2019.08.017","DOIUrl":null,"url":null,"abstract":"<div><p>Gallai (1966) conjectured that the edge set of every graph <em>G</em> on <em>n</em> vertices can be covered by at most ⌈<em>n</em>/2⌉ edge-disjoint paths. Such a covering by edge-disjoint paths is called a <em>path decomposition</em>, and the size of a path decomposition with a minimum number of elements is called the <em>path number</em> of <em>G</em>. Peroche (1984) proved that the problem of computing the path number is NP-Complete; and Constantinou and Ellinas (2018) proved that it is polynomial for a family of complete bipartite graphs. In this paper we present an Integer Linear Programming model for computing the path number of a graph. This allowed us to verify Gallai's Conjecture for a large collection of graphs. As a result, following a work of Heinrich, Natale and Streicher on cycle decompositions (2017), we verify Gallai's Conjecture for graphs with at most 11 vertices; for bipartite graphs with at most 16 vertices; and for regular graphs with at most 14 vertices.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":"346 ","pages":"Pages 185-197"},"PeriodicalIF":0.0000,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.017","citationCount":"1","resultStr":"{\"title\":\"On Computing the Path Number of a Graph\",\"authors\":\"F. Botler, R. Cano, M. Sambinelli\",\"doi\":\"10.1016/j.entcs.2019.08.017\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Gallai (1966) conjectured that the edge set of every graph <em>G</em> on <em>n</em> vertices can be covered by at most ⌈<em>n</em>/2⌉ edge-disjoint paths. Such a covering by edge-disjoint paths is called a <em>path decomposition</em>, and the size of a path decomposition with a minimum number of elements is called the <em>path number</em> of <em>G</em>. Peroche (1984) proved that the problem of computing the path number is NP-Complete; and Constantinou and Ellinas (2018) proved that it is polynomial for a family of complete bipartite graphs. In this paper we present an Integer Linear Programming model for computing the path number of a graph. This allowed us to verify Gallai's Conjecture for a large collection of graphs. As a result, following a work of Heinrich, Natale and Streicher on cycle decompositions (2017), we verify Gallai's Conjecture for graphs with at most 11 vertices; for bipartite graphs with at most 16 vertices; and for regular graphs with at most 14 vertices.</p></div>\",\"PeriodicalId\":38770,\"journal\":{\"name\":\"Electronic Notes in Theoretical Computer Science\",\"volume\":\"346 \",\"pages\":\"Pages 185-197\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-08-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.017\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Notes in Theoretical Computer Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1571066119300672\",\"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/S1571066119300672","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Computer Science","Score":null,"Total":0}
Gallai (1966) conjectured that the edge set of every graph G on n vertices can be covered by at most ⌈n/2⌉ edge-disjoint paths. Such a covering by edge-disjoint paths is called a path decomposition, and the size of a path decomposition with a minimum number of elements is called the path number of G. Peroche (1984) proved that the problem of computing the path number is NP-Complete; and Constantinou and Ellinas (2018) proved that it is polynomial for a family of complete bipartite graphs. In this paper we present an Integer Linear Programming model for computing the path number of a graph. This allowed us to verify Gallai's Conjecture for a large collection of graphs. As a result, following a work of Heinrich, Natale and Streicher on cycle decompositions (2017), we verify Gallai's Conjecture for graphs with at most 11 vertices; for bipartite graphs with at most 16 vertices; and for regular graphs with at most 14 vertices.
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