  Heat flow in nanomaterials is an important area of study, with both
fundamental and technological implications. However, little is known about heat
flow in two-dimensional (2D) devices or interconnects with dimensions
comparable to the phonon mean free path (mfp). Here, we find that short,
quarter-micron graphene samples reach ~35% of the ballistic heat conductance
limit up to room temperature, enabled by the relatively large phonon mfp (~100
nm) in substrate-supported graphene. In contrast, patterning similar samples
into nanoribbons (GNRs) leads to a diffusive heat flow regime that is
controlled by ribbon width and edge disorder. In the edge-controlled regime,
the GNR thermal conductivity scales with width approximately as ~W^{1.8+/-0.3},
being about 100 W/m/K in 65-nm-wide GNRs, at room temperature. Manipulation of
device dimensions on the scale of the phonon mfp can be used to achieve full
control of their heat-carrying properties, approaching fundamentally limited
upper or lower bounds.
