Covering matroid

In this paper, we propose a new type of matroids, namely covering matroids, and investigate the connections with the second type of covering-based rough sets and some existing special matroids. Firstly, as an extension of partitions, coverings are more natural combinatorial objects and can sometimes be more efficient to deal with problems in the real world. Through extending partitions to coverings, we propose a new type of matroids called covering matroids and prove them to be an extension of partition matroids. Secondly, since some researchers have successfully applied partition matroids to classical rough sets, we study the relationships between covering matroids and covering-based rough sets which are an extension of classical rough sets. Thirdly, in matroid theory, there are many special matroids, such as transversal matroids, partition matroids, 2-circuit matroid and partition-circuit matroids. The relationships among several special matroids and covering matroids are studied.Comment: 15 page

Hopf algebras for matroids over hyperfields

Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields

Multi-Path Matroids

We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural properties. Key elements of this work are two complementary perspectives we develop for these matroids: on the one hand, multi-path matroids are transversal matroids that have special types of presentations; on the other hand, the bases of multi-path matroids can be viewed as sets of lattice paths in certain planar diagrams.Comment: 24 pages; 10 figure