Visibility Graphs of Staircase Polygons and the Weak Bruhat Order, I: from Visibility Graphs to Maximal Chains

The recognition problem for visibility graphs of simple polygons is not known to be in NP, nor is it known to be NP-hard. It is, however, known to be in PSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans. Any nondegenerate configuration of n points can be associated with a maximal chain in the weak Bruhat order of the symmetric group S n. The visibility graph of any simple polygon defined on this configuration is completely determined by this maximal chain via a one-to-one correspondence between maximal chains and balanced tableaux of a certain shape. In the case of staircase polygons (special convex fans), we define a class of graphs called persistent graphs and show that the visibility graph of a staircase polygon is persistent. We then describe a polynomial-time algorithm that recovers a representative maximal chain in the weak Bruhat order from a given persistent graph, thus characterizing the class of persistent graphs. The question of recovering a staircase polygon from a given persistent graph, via a maximal chain, is studied in the companion paper [4]. The overall goal of both papers is to offer a characterization of visibility graphs of convex fans. * The research of J. Abello was supported by NSF Grants Nos. DCR 8603722 and DCR 8896281. This research was done while K. Kumar was at the Department of Computer Science, Texas A & M University. 332 J. Abello, O. Egecioglu, and K. Kumar