  We investigate a spin-$1/2$ two-leg honeycomb ladder with frustrating
next-nearest-neighbor (NNN) coupling along the legs, which is equivalent to two
$J_1$-$J_2$ spin chains coupled with $J_\perp$ at odd rungs. The full parameter
region of the model is systematically studied using conventional and infinite
density-matrix renormalization group as well as bosonization. The rich phase
diagram consists of five distinct phases: A Haldane phase, a NNN-Haldane phase
and a staggered dimer phase when $J_{\perp} < 0$; a rung singlet phase and a
columnar dimer phase when $J_{\perp} > 0$. An interesting reentrant behavior
from the dimerized phase into the Haldane phase is found as the frustration
$J_2$ increases. The universalities of the critical phase transitions are fully
analyzed. Phase transitions between dimerized and disordered phases belong to
the two-dimensional Ising class with central charge $c=1/2$. The transition
from the Haldane phase to NNN-Haldane phase is of a weak topological first
order, while the continuous transition between the Haldane phase and rung
singlet phase has central charge $c=2$.
