A clutter is a family of sets, called members, such that no member contains another. It is called intersecting if every two members intersect, but not all members have a common element. Dense clutters additionally do not have a fractional packing of value 2. We are looking at certain substructures of clutters, namely minors and restrictions.

For a family of clutters we introduce a general sufficient condition such that for every clutter we can decide whether the clutter has a restriction in that set in polynomial time. It is known that the sets of intersecting and dense clutters satisfy this condition.
For intersecting clutters we generalize the statement to k-wise intersecting clutters using a much simpler proof.

We also give a simplified proof that a dense clutter with no proper dense minor is either a delta or the blocker of an extended odd hole. This simplification reduces the running time of the algorithm for finding a delta or the blocker of an extended odd hole minor from previously $O(n^4)$ to $O(n^3)$ filter oracle calls.

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