B0_0-VPG Representation of AT-free Outerplanar Graphs

B$_0$-VPG graphs are intersection graphs of axis-parallel line segments in the plane. In this paper, we show that all AT-free outerplanar graphs are B$_0$-VPG. We first prove that every AT-free outerplanar graph is an induced subgraph of a biconnected outerpath (biconnected outerplanar graphs whose weak dual is a path) and then we design a B$_0$-VPG drawing procedure for biconnected outerpaths. Our proofs are constructive and give a polynomial time B$_0$-VPG drawing algorithm for the class. We also characterize all subgraphs of biconnected outerpaths and name this graph class "linear outerplanar". This class is a proper superclass of AT-free outerplanar graphs and a proper subclass of outerplanar graphs with pathwidth at most 2. It turns out that every graph in this class can be realized both as an induced subgraph and as a spanning subgraph of (different) biconnected outerpaths.Comment: A preliminary version, which did not contain the characterization of linear outerplanar graphs (Section 3), was presented in the $8^{th}$ International Conference on Algorithms and Discrete Applied Mathematics (CALDAM) 2022. The definition of linear outerplanar graphs in this paper differs from that in the preliminary version and hence Section 4 is ne