Boundary Properties of Factorial Classes of Graphs
For a class of graphs X, let Xn be the number of graphs with vertex set {1,…,n} in the class X, also known as the speed of X. It is known that in the family of hereditary classes (i.e. those that are closed under taking induced subgraphs) the speeds constitute discrete layers and the first four lowe...
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Published in | Journal of graph theory Vol. 78; no. 3; pp. 207 - 218 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Hoboken
Blackwell Publishing Ltd
01.03.2015
Wiley Subscription Services, Inc |
Subjects | |
Online Access | Get full text |
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Summary: | For a class of graphs X, let Xn be the number of graphs with vertex set {1,…,n} in the class X, also known as the speed of X. It is known that in the family of hereditary classes (i.e. those that are closed under taking induced subgraphs) the speeds constitute discrete layers and the first four lower layers are constant, polynomial, exponential, and factorial. For each of these four layers a complete list of minimal classes is available, and this information allows to provide a global structural characterization for the first three of them. The minimal layer for which no such characterization is known is the factorial one. A possible approach to obtaining such a characterization could be through identifying all minimal superfactorial classes. However, no such class is known and possibly no such class exists. To overcome this difficulty, we employ the notion of boundary classes that has been recently introduced to study algorithmic graph problems and reveal the first few boundary classes for the factorial layer. |
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Bibliography: | Centre for Discrete Mathematics and Its Applications (DIMAP), University of Warwick Russian Federation President - No. MK-1148.2013.1 istex:E1D19F832B2A4DDE19443C1DB5596A8ED6B88570 ark:/67375/WNG-B3D0XV9V-K Russian Federation Government - No. 11.G34.31.0057 National Research University - Higher School of Economics' Academic Fund Program in 2014/2015 - No. 14-01-0002 ArticleID:JGT21799 EPSRC - No. EP/I01795X/1 Research of Vadim Lozin was supported by the Centre for Discrete Mathematics and Its Applications (DIMAP), University of Warwick and by EPSRC, grant EP/I01795X/1. Research of Victor Zamaraev was supported by The National Research University – Higher School of Economics' Academic Fund Program in 2014/2015 (research grant No 14‐01‐0002), by Russian Federation President Grant MK‐1148.2013.1 and by Russian Federation Government grant N. 11.G34.31.0057. |
ISSN: | 0364-9024 1097-0118 |
DOI: | 10.1002/jgt.21799 |