Before revision:  We establish lower and upper bounds on the new variants of the segment number, mostly for cubic graphs, depending on the connectivity of the given graph. We also construct an infinite family of planar graphs where the classical segment number is asymptotically twice as large as each of the new variants of the segment number  .
After revision:  We first construct an infinite family of planar graphs where the classical segment number is asymptotically twice as large as each of the new variants of the segment number .