  We define and study a class of graphs, called 2-stab interval graphs (2SIG),
with boxicity 2 which properly contains the class of interval graphs. A 2SIG is
an axes-parallel rectangle intersection graph where the rectangles have unit
height (that is, length of the side parallel to $Y$-axis) and intersects either
of the two fixed lines, parallel to the $X$-axis, distance $1+\epsilon$ ($0 <
\epsilon < 1$) apart. Intuitively, 2SIG is a graph obtained by putting some
edges between two interval graphs in a particular rule. It turns out that for
these kind of graphs, the chromatic number of any of its induced subgraphs is
bounded by twice of its (induced subgraph) clique number. This shows that the
graph, even though not perfect, is not very far from it. Then we prove similar
results for some subclasses of 2SIG and provide efficient algorithm for finding
their clique number. We provide a matrix characterization for a subclass of
2SIG graph.
