Michael Formann, T. Hagerup, J. Haralambides, Michael Kaufmann, F. Leighton, A. Symvonis, E. Welzl, G. Woeginger
{"title":"在平面上绘制高分辨率的图形","authors":"Michael Formann, T. Hagerup, J. Haralambides, Michael Kaufmann, F. Leighton, A. Symvonis, E. Welzl, G. Woeginger","doi":"10.1109/FSCS.1990.89527","DOIUrl":null,"url":null,"abstract":"The problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized is studied. The resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph, and the resolution of a graph is defined to be the maximum resolution of any layout of the graph. The resolution R of a graph is characterized in terms of the maximum node degree d of the graph by proving that Omega (1/d/sup 2/)<or=R<or=2 pi /d for any graph. Moreover, it is proved that R= Theta (1/d) for many graphs, including planar graphs, complete graphs, hypercubes, multidimensional meshes and tori, and other special networks. It is also shown that the problem of deciding if R=2 pi /d for a graph is NP-hard for d=4, and a counting argument is used to show that R=O(log d/d/sup 2/) for many graphs.<<ETX>>","PeriodicalId":271949,"journal":{"name":"Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science","volume":"13 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1990-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"24","resultStr":"{\"title\":\"Drawing graphs in the plane with high resolution\",\"authors\":\"Michael Formann, T. Hagerup, J. Haralambides, Michael Kaufmann, F. Leighton, A. Symvonis, E. Welzl, G. Woeginger\",\"doi\":\"10.1109/FSCS.1990.89527\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized is studied. The resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph, and the resolution of a graph is defined to be the maximum resolution of any layout of the graph. The resolution R of a graph is characterized in terms of the maximum node degree d of the graph by proving that Omega (1/d/sup 2/)<or=R<or=2 pi /d for any graph. Moreover, it is proved that R= Theta (1/d) for many graphs, including planar graphs, complete graphs, hypercubes, multidimensional meshes and tori, and other special networks. It is also shown that the problem of deciding if R=2 pi /d for a graph is NP-hard for d=4, and a counting argument is used to show that R=O(log d/d/sup 2/) for many graphs.<<ETX>>\",\"PeriodicalId\":271949,\"journal\":{\"name\":\"Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science\",\"volume\":\"13 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1990-10-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"24\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/FSCS.1990.89527\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/FSCS.1990.89527","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized is studied. The resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph, and the resolution of a graph is defined to be the maximum resolution of any layout of the graph. The resolution R of a graph is characterized in terms of the maximum node degree d of the graph by proving that Omega (1/d/sup 2/)>
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