Graphs of bounded chordality

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.