I am interested in the problem of covering (rectilinear) regions with rectangles that we can start to formulate in graph-theoretic terms as follows:
Consider a region $V \subseteq S$ where $S$ is the set of squares from a two-dimensional grid. A rectangle of $V$ is a subset of $V$ whose union is a rectangle on the plane. Then $G(V, E)$ is a graph whose vertices are squares from $V$ and there is an edge between two vertices (squares) if there is a rectangle of $V$ that contains them both.
Now I wonder which class of graphs such graphs belong to. Chordal graphs seem fitting and I couldn't generate a counter-example so far but is it true? Are such graphs chordal graphs?
EDIT: The answer is no and there is a simple counter-example (see Alex's answer below). Therefore I ask the same question by requiring that the region $V$ is without holes.
If $V$ may contain holes then there is a simple counterexample:
Here 1-2-3-4-1 is an elementary chordless cycle of length four in $G(V,E)$.
Otherwise there still is a counterexample:
So there remains a following converse conjecture. If the graph $G(V,E)$ is chordal then the set $V$ contains no hole $H$ surrounded by squares from $V$ (the latter means that each square from $S$ which has a common boundary point with $H$ belongs to $V$). I guess that we can try to prove the conjecture by showing that an elementary cycle of minimal length “around” the hole $H$ is chordless.
On the other hand, maybe the graph $G(V,E)$ is chordal provided $V$ is convex wrt. grid directions, that is if $s$ and $s’$ are squares from $V$ such that a segment connecting their centers is parallel to a grid line then all squares from $S$ between $s$ and $s’$ also belong to $V$.