  The segment number of a planar graph $G$ is the smallest number of line
segments needed for a planar straight-line drawing of $G$. Dujmovi\'c,
Eppstein, Suderman, and Wood [CGTA'07] introduced this measure for the visual
complexity of graphs. There are optimal algorithms for trees and worst-case
optimal algorithms for outerplanar graphs, 2-trees, and planar 3-trees. It is
known that every cubic triconnected planar $n$-vertex graph (except $K_4$) has
segment number $n/2+3$, which is the only known universal lower bound for a
meaningful class of planar graphs.
  We show that every triconnected planar 4-regular graph can be drawn using at
most $n+3$ segments. This bound is tight up to an additive constant, improves a
previous upper bound of $7n/4+2$ implied by a more general result of Dujmovi\'c
et al., and supplements the result for cubic graphs. We also give a simple
optimal algorithm for cactus graphs, generalizing the above-mentioned result
for trees. We prove the first linear universal lower bounds for outerpaths,
maximal outerplanar graphs, 2-trees, and planar 3-trees. This shows that the
existing algorithms for these graph classes are constant-factor approximations.
For maximal outerpaths, our bound is best possible and can be generalized to
circular arcs.
