A generalisation of the matroid lift construction
Geoff Whittle · Transactions of the American Mathematical Society · 1989
This paper introduces a general matroid-theoretic construction which includes, as special cases, elementary lifts of matroids and bias matroids of biased graphs. To perform the construction on a matroid M M , it is necessary (but not sufficient) to have a submodular function inducing M M . Elementary lifts are obtained when the submodular function chosen is the rank function of M M . We define what is meant by a k k - induced matroid. These matroids simultaneously generalise matroids of graphs, transversal matroids and Dilworth truncations. They are induced by a particularly natural class of submodular functions. The effect of the above construction on k k -induced matroids using these natural submodular functions is studied. Results on minors of k k -induced matroids and the matroids obtained from them using the construction are given.