Foto Afrati, Christos H. Papadimitriou , George Papageorgiou
{"title":"三次图的复杂性","authors":"Foto Afrati, Christos H. Papadimitriou , George Papageorgiou","doi":"10.1016/S0019-9958(85)80012-7","DOIUrl":null,"url":null,"abstract":"<div><p>A graph is cubical if it is a subgraph of a hypercube; the dimension of the smallest such hypercube is the dimension of the graph. We show several results concerning this class of graphs. We use a characterization of cubical graphs in terms of edge coloring to show that the dimension of biconnected cubical graphs is at most half the number of nodes. We also show that telling whether a graph is cubical is NP-complete. Finally, we propose a heuristic for minimizing the dimension of trees, which yields an embedding of the tree in a hypercube of dimension at most the square of the true dimension of the tree.</p></div>","PeriodicalId":38164,"journal":{"name":"信息与控制","volume":"66 1","pages":"Pages 53-60"},"PeriodicalIF":0.0000,"publicationDate":"1985-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0019-9958(85)80012-7","citationCount":"23","resultStr":"{\"title\":\"The complexity of cubical graphs\",\"authors\":\"Foto Afrati, Christos H. Papadimitriou , George Papageorgiou\",\"doi\":\"10.1016/S0019-9958(85)80012-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A graph is cubical if it is a subgraph of a hypercube; the dimension of the smallest such hypercube is the dimension of the graph. We show several results concerning this class of graphs. We use a characterization of cubical graphs in terms of edge coloring to show that the dimension of biconnected cubical graphs is at most half the number of nodes. We also show that telling whether a graph is cubical is NP-complete. Finally, we propose a heuristic for minimizing the dimension of trees, which yields an embedding of the tree in a hypercube of dimension at most the square of the true dimension of the tree.</p></div>\",\"PeriodicalId\":38164,\"journal\":{\"name\":\"信息与控制\",\"volume\":\"66 1\",\"pages\":\"Pages 53-60\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1985-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0019-9958(85)80012-7\",\"citationCount\":\"23\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"信息与控制\",\"FirstCategoryId\":\"1093\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019995885800127\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"信息与控制","FirstCategoryId":"1093","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019995885800127","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
A graph is cubical if it is a subgraph of a hypercube; the dimension of the smallest such hypercube is the dimension of the graph. We show several results concerning this class of graphs. We use a characterization of cubical graphs in terms of edge coloring to show that the dimension of biconnected cubical graphs is at most half the number of nodes. We also show that telling whether a graph is cubical is NP-complete. Finally, we propose a heuristic for minimizing the dimension of trees, which yields an embedding of the tree in a hypercube of dimension at most the square of the true dimension of the tree.
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