  A hole in a graph is an induced subgraph which is a cycle of length at least
four. A graph is chordal if it contains no holes. Following McKee and
Scheinerman (1993), we define the chordality of a graph $G$ to be the minimum
number of chordal graphs on $V(G)$ such that the intersection of their edge
sets is equal to $E(G)$. In this paper we study classes of graphs of bounded
chordality.
  In the 1970s, Buneman, Gavril, and Walter, proved independently that chordal
graphs are exactly the intersection graphs of subtrees in trees. We generalize
this result by proving that the graphs of chordality at most $k$ are exactly
the intersection graphs of convex subgraphs of median graphs of tree-dimension
$k$.
  A hereditary class of graphs $\mathcal{A}$ is $\chi$-bounded if there exists
a function $ f\colon \mathbb{N}\rightarrow \mathbb{R}$ such that for every
graph $G\in \mathcal{A}$, we have $\chi(G) \leq f(\omega(G))$. In 1960, Asplund
and Gr\"unbaum proved that the class of all graphs of boxicity at most two is
$\chi$-bounded. In his seminal paper "Problems from the world surrounding
perfect graphs," Gy\'arf\'as (1985), motivated by the above result, asked
whether the class of all graphs of chordality at most two, which we denote by
$\mathcal{C}\gcap \mathcal{C}$, is $\chi$-bounded. We discuss a result of
Felsner, Joret, Micek, Trotter and Wiechert (2017), concerning
tree-decompositions of Burling graphs, which implies an answer to Gy\'arf\'as'
question in the negative. We prove that two natural families of subclasses of
$\mathcal{C}\gcap \mathcal{C}$ are polynomially $\chi$-bounded.
  Finally, we prove that for every $k\geq 3$ the $k$-\textsc{Chordality
Problem}, which asks to decide whether a graph has chordality at most $k$, is
\textbf{NP}-complete.
