Let Γ be a collection of unbounded x-monotone Jordan arcs intersecting at most twice each other, which we call pseudo-parabolas, since two axis parallel parabolas intersects at most twice. We investigate how to cut pseudo-parabolas into the minimum number of curve segments so that each pair of segments intersect at most once. We give an Ω(n4/3) lower bound and O(n5/3) upper bound. We give the same bounds for an arrangement of circles. Applying the upper bound, we give an O(n23/12) bound on the complexity of a level of pseudo-parabolas, and O(n11/6) bound on the complexity of a combinatorially concave chain of pseudo parabolas. We also give some upperbounds on the number of transitions of the minimum weight matroid base when the weight of each element changes as a quadratic function of a single parameter.