  A $k$-bend path is a non-self-intersecting polyline in the plane made of at
most $k+1$ axis-parallel line segments. B$_k$-VPG is the class of graphs which
can be represented as intersection graphs of $k$-bend paths in the same plane.
In this paper, we show that all AT-free outerplanar graphs are B$_0$-VPG, i.e.,
intersection graphs of horizontal and vertical line segments in the plane. Our
proofs are constructive and give a polynomial time B$_0$-VPG drawing algorithm
for the class.
  Following a long line of improvements, Gon\c{c}alves, Isenmann, and Pennarun
[SODA 2018] showed that all planar graphs are B$_1$-VPG. Since there are planar
graphs which are not B$_0$-VPG, characterizing B$_0$-VPG graphs among planar
graphs becomes interesting. Chaplick et al.\ [WG 2012] had shown that it is
NP-complete to recognize B$_k$-VPG graphs within B$_{k+1}$-VPG. Hence
recognizing B$_0$-VPG graphs within B$_1$-VPG is NP-complete in general, but
the question is open when restricted to planar graphs. There are outerplanar
graphs and AT-free planar graphs which are not B$_0$-VPG. This piqued our
interest in AT-free outerplanar graphs.
