Matroid Theory¶
Oxley, J. (2011). Matroid Theory. Oxford University Press.
Cited by¶
3 citations across 3 artifacts.
Each citation links to the sentence it supports in the citing article.
Domain-specific¶
- Graph Duality
- A proper vertex \(k\)-coloring of \(G^*\) corresponds to a proper face \(k\)-coloring of \(G\), making the four-color theorem equivalently a statement about face colorings in the dual. Spanning trees of \(G\) correspond bijectively to co-trees (complements of spanning trees) in \(G^*\), a duality that organizes much of planar matroid theory
This sourceMatroid duality and the duals of graphic matroids, the setting in which planar duality pairs spanning trees with co-trees.
Supported in partVerified against the source
- A proper vertex \(k\)-coloring of \(G^*\) corresponds to a proper face \(k\)-coloring of \(G\), making the four-color theorem equivalently a statement about face colorings in the dual. Spanning trees of \(G\) correspond bijectively to co-trees (complements of spanning trees) in \(G^*\), a duality that organizes much of planar matroid theory
- Independence system
Mechanisms¶
- Dependency Elimination Test
- Its subtle failure is order dependence: a set can admit several different minimal subsets, so which generators survive depends on the order you tested them, and two honest runs can disagree about the "right" primitives even when they agree on the count.
This sourceShows that a matroid may have multiple bases of equal size and that greedy tie or order choices can select different bases.
- Its subtle failure is order dependence: a set can admit several different minimal subsets, so which generators survive depends on the order you tested them, and two honest runs can disagree about the "right" primitives even when they agree on the count.
Verification¶
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