Grounded L-Graphs Are Polynomially X-Bounded

A grounded L-graph is the intersection graph of a collection of "L" shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number omega has chromatic number at most 17 omega(4). This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially x-bounded. We also survey x-boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.