  A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square
intersection graph where the unit squares intersect either of the two fixed
lines parallel to the $X$-axis, distance $1 + \epsilon$ ($0 < \epsilon < 1$)
apart. This family of graphs allow us to study local structures of unit square
intersection graphs, that is, graphs with cubicity 2. The complexity of
determining whether a tree has cubicity 2 is unknown while the graph
recognition problem for unit square intersection graph is known to be NP-hard.
We present a polynomial time algorithm for recognizing trees that admit a 2SUIG
representation.
