Matchings and \(\Delta\)-matroids
[This is a joint review for the paper mentioned above [Zbl 0708.05013] (first paper) and this one (second paper).] In these important papers the author introduces a very interesting structure and proves a lot of beautiful results. Let \(\Delta\) denote symmetric difference and replace \(\setminus\) by \(\Delta\) in the matroid base axiom ``For X,Y\(\in {\mathcal F}\) and \(x\in X\setminus Y\) there exists a \(y\in Y\setminus X\) such that \(X\Delta\{\) x,y\(\}\in {\mathcal F}.''\) Hence a generalization of matroids, called \(\Delta\)-matroids, is introduced. The first paper presents results on representing \(\Delta\)-matroids by antisymmetric matrices, on their relations with Eulerian tours of 4- regular graphs, with matroids introduced by Dress and Havel, and with electrically self-dual matroids, introduced by the reviewer. The second paper characterizes \(\Delta\)-matroids via a greedy algorithm, studies the relations between matchings and \(\Delta\)-matroids, and defines the union of \(\Delta\)-matroids and the induction of \(\Delta\)-matroids by a bipartite graph.
scientific article; zbMATH DE number 4162894