I came up with this problem yesterday: Let $R$ be a rectangle in $\mathbb{R}^2$, $P$ a finite partition of $R$ into rectangles such that every one of them has its sides parallel to the sides of $R$ and $P_c$ the set of the centers of the rectangles in $P$. Construct a graph as follows: Draw a segment between any two different $r_c,r_c'\in P_c$ iff their respective rectangles share more than a vertex. We call this graph (for instance) The Right Graph of $R$ (haven't actually looked if there is already a name for such graph), and denote it as $G_R$.
Given a rectangle $R$, finding its right graph is not that complicated, so my question is, what are the conditions for a graph $G$ to be the right graph of a rectangle in $\mathbb{R}^2$ ??