The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e. when the goal is to decide if two common independent sets suffice or not. Nevertheless, as the problem generalizes several long-standing open questions, identifying tractable cases is of particular interest. Strongly base orderable matroids form a class for which a basis-exchange condition that is much stronger than the standard axiom is met. As a result, several problems that are open for arbitrary matroids can be solved for this class. In particular, Davies and McDiarmid showed that if both matroids are strongly base orderable, then the covering number of their intersection coincides with the maximum of their covering numbers.

Motivated by their result, we propose relaxations of strongly base orderability in two directions. First we weaken the basis-exchange condition, which leads to the definition of a new, complete class of matroids with distinguished algorithmic properties. Second, we introduce the notion of covering the circuits of a matroid by a graph, and consider the cases when the graph is (A) 2-regular, or (B) a path. We give an extensive list of results explaining how the proposed relaxations compare to existing conjectures and theorems on coverings by common independent sets.

即使在非常有限的设置中，即当目标是确定两个公共独立集是否足够时，用最小数量的公共独立集覆盖两个拟阵的地面集的问题也是众所周知的困难。然而，由于该问题概括了几个长期存在的悬而未决的问题，因此识别可处理的案例特别令人感兴趣。强基可有序拟阵形成一类，其满足比标准公理强得多的基交换条件。因此，此类可以解决任意拟阵开放的几个问题。特别是，Davies 和 McDiarmid 证明，如果两个拟阵都是强碱可序的，那么它们的交集的覆盖数就与它们的覆盖数的最大值一致。

受他们结果的启发，我们提出在两个方向上放松强基有序性。首先，我们弱化了基交换条件，从而定义了一个新的、完整的、具有杰出算法特性的拟阵类。其次，我们引入了用图覆盖拟阵电路的概念，并考虑图是（A）2-正则图或（B）路径的情况。我们给出了广泛的结果列表，解释了所提出的松弛与关于公共独立集覆盖的现有猜想和定理的比较。