A Graphical Representation of Matroids
A base graph of a matroid is the graph whose points are the bases of the matroid. Two bases are adjacent if they differ by exactly one element. A definition of equivalence of matroids is given and it is shown that two matroids are equivalent if and only if their base graphs are isomorphic. In partic...
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| Veröffentlicht in: | SIAM journal on applied mathematics 1973-12, Vol.25 (4), p.618-627 |
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| container_end_page | 627 |
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| container_title | SIAM journal on applied mathematics |
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| creator | Holzmann, C. A. Norton, P. G. Tobey, M. D. |
| description | A base graph of a matroid is the graph whose points are the bases of the matroid. Two bases are adjacent if they differ by exactly one element. A definition of equivalence of matroids is given and it is shown that two matroids are equivalent if and only if their base graphs are isomorphic. In particular, if M and M1 are nonseparable matroids with isomorphic base graphs, then M is isomorphic to either M1 or its dual. Thus, the study of matroids is reduced to the study of a class of graphs: the base graphs. A detailed investigation of the structure of neighborhoods in the base graph is carried out and this is used to establish the above result. Included in the result is a graphical classification of separable matroids which gives a new proof of Whitney's decomposition theorem. |
| doi_str_mv | 10.1137/0125060 |
| format | Article |
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A.</au><au>Norton, P. G.</au><au>Tobey, M. D.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Graphical Representation of Matroids</atitle><jtitle>SIAM journal on applied mathematics</jtitle><date>1973-12-01</date><risdate>1973</risdate><volume>25</volume><issue>4</issue><spage>618</spage><epage>627</epage><pages>618-627</pages><issn>0036-1399</issn><eissn>1095-712X</eissn><abstract>A base graph of a matroid is the graph whose points are the bases of the matroid. Two bases are adjacent if they differ by exactly one element. A definition of equivalence of matroids is given and it is shown that two matroids are equivalent if and only if their base graphs are isomorphic. In particular, if M and M1 are nonseparable matroids with isomorphic base graphs, then M is isomorphic to either M1 or its dual. Thus, the study of matroids is reduced to the study of a class of graphs: the base graphs. 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| fulltext | fulltext |
| identifier | ISSN: 0036-1399 |
| ispartof | SIAM journal on applied mathematics, 1973-12, Vol.25 (4), p.618-627 |
| issn | 0036-1399 1095-712X |
| language | eng |
| recordid | cdi_proquest_journals_917622450 |
| source | JSTOR Mathematics & Statistics; Jstor Complete Legacy; LOCUS - SIAM's Online Journal Archive |
| subjects | Decomposition Graph theory Graphics Graphs Induced subgraphs Line graphs Mathematics Matroids Neighborhoods |
| title | A Graphical Representation of Matroids |
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