  We present an approximation algorithm for the maximum independent set (MIS)
problem over the class of equilateral $B_1$-VPG graphs. These are intersection
graphs of $L$-shaped planar objects % (and their rotations by multiples of
$90^o$) with both arms of each object being equal. We obtain a $36(\log
2d)$-approximate algorithm running in $O(n(\log n)^2)$ time for this problem,
where $d$ is the ratio $d_{max}/d_{min}$ and $d_{max}$ and $d_{min}$ denote
respectively the maximum and minimum length of any arm in the input equilateral
$L$-representation of the graph. In particular, we obtain $O(1)$-factor
approximation of MIS for $B_1$-VPG -graphs for which the ratio $d$ is bounded
by a constant. % formed by unit length $L$-shapes. In fact, algorithm can be
generalized to an $O(n(\log n)^2)$ time and a $36(\log 2d_x)(\log
2d_y)$-approximate MIS algorithm over arbitrary $B_1$-VPG graphs. Here, $d_x$
and $d_y$ denote respectively the analogues of $d$ when restricted to only
horizontal and vertical arms of members of the input. This is an improvement
over the previously best $n^\epsilon$-approximate algorithm \cite{FoxP} (for
some fixed $\epsilon>0$), unless the ratio $d$ is exponentially large in $n$.
In particular, $O(1)$-approximation of MIS is achieved for graphs with
$\max\{d_x,d_y\}=O(1)$.
