Intertwining connectivities in representable matroids
Let $M$ be a representable matroid and $Q, R, S, T$ subsets of the ground set such that the smallest separation that separates $Q$ from $R$ has order $k$, and the smallest separation that separates $S$ from $T$ has order $l$. We prove that if $M$ is sufficiently large, then there is an element $e$ such that in one of $M\backslash e$ and $M\!/e$ both connectivities are preserved. For matroids representable over a finite field we prove a stronger result: we show that we can remove $e$ such that both a connectivity and a minor of $M$ are preserved.
|Keywords||matroids, connectivity, Tutte's linking theorem, interwining|
|THEME||Logistics (theme 3)|
|Journal||SIAM Journal on Discrete Mathematics|
|Project||Matroid Structure for Efficiency|
Huynh, T.C.T, & van Zwam, S.H.M. (2014). Intertwining connectivities in representable matroids. SIAM Journal on Discrete Mathematics, 28(1), 188–196. doi:10.1137/13091837X