Characterization and a 2D Visualization of B${}_0$-VPG Cocomparability Graphs

B${}_0$-VPG graphs are intersection graphs of vertical and horizontal line segments on a plane. Cohen, Golumbic, Trotter, and Wang [Order, 2016] pose the question of characterizing B${}_0$-VPG permutation graphs. We respond here by characterizing B${}_0$-VPG cocomparability graphs. This characterization also leads to a polynomial time recognition and B${}_0$-VPG drawing algorithm for the class. Our B${}_0$-VPG drawing algorithm starts by fixing any one of the many posets $P$ whose cocomparability graph is the input graph $G$. The drawing we obtain not only visualizes $G$ in that one can distinguish comparable pairs from incomparable ones, but one can also identify which among a comparable pair is larger in $P$ from this visualization.