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 $17omega^4$. This improves
the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially $chi$-bounded. We also survey $chi$-boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number $omega$ has chromatic number at most $3cdot 4^{omega-1}
$. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time.
Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was
conjectured by Hlinv{e}ny, Kwon, Obdrv{z}alek, and Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer $t$, the class of graphs with no vertex-minor isomorphic to the path on $t$ vertices has bounded shrub-depth.
