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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 inPSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans.  Any nondegenerated configuration ofn points can be associated with amaximal chain in the weak Bruhat order of the symmetric groupS  n . The visibility graph ofany simple polygon defined on this configuration is completely determined by this maximal chain via a one-to-one correspondence between maximal chains andbalanced tableaux of a certain shape.    In the case of staircase polygons (special convex fans), we define a class of graphs calledpersistent 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.

Visibility graphs of staircase polygons and the weak Bruhat order, I: From visibility graphs to maximal chains