  We use Quantum Monte-Carlo methods to study the ground state phase diagram of
a S=1/2 honeycomb lattice magnet in which a nearest-neighbor antiferromagnetic
exchange J (favoring N\'eel order) competes with two different multi-spin
interaction terms: a six-spin interaction Q_3 that favors columnar valence-bond
solid (VBS) order, and a four-spin interaction Q_2 that favors staggered VBS
order. For Q_3 ~ Q_2 >> J, we establish that the competition between the two
different VBS orders stabilizes N\'eel order in a large swathe of the phase
diagram even when J is the smallest energy-scale in the Hamiltonian. When Q_3
>> (Q_2,J) (Q_2 >> (Q_3,J)), this model exhibits at zero temperature phase
transition from the N\'eel state to a columnar (staggered) VBS state. We
establish that the N\'eel-columnar VBS transition is continuous for all values
of Q_2, and that critical properties along the entire phase boundary are
well-characterized by critical exponents and amplitudes of the non-compact CP^1
(NCCP^1) theory of deconfined criticality, similar to what is observed on a
square lattice. However, a surprising three-fold anisotropy of the phase of the
VBS order parameter at criticality, whose presence was recently noted at the
Q_2=0 deconfined critical point, is seen to persist all along this phase
boundary. We use a classical analogy to explore this by studying the critical
point of a three-dimensional XY model with a four-fold anisotropy field which
is known to be weakly irrelevant at the three-dimensional XY critical point. In
this case, we again find that the critical anisotropy appears to saturate to a
nonzero value over the range of sizes accessible to our simulations.
