This question concerns the structure of a directed graph built on the cells of an arrangement of lines. My basic question is whether this graph has been studied before, perhaps in another guise. I also ask a specific question.

Let $\cal A$ be an arrangement of $n$ lines in $\mathbb{R}^2$, with no line vertical, and no two lines parallel. I associate to $\cal A$ a directed graph $G$, with each node a cell of $\cal A$, and $a \to b$ iff cell $a$ is adjacently above $b$ in the sense that (1) $a$ and $b$ share an edge $e$, and (2) for a point $p$ in the interior of $e$, points vertically just below $p$ lie in $b$, and points vertically just above $p$ lie in $a$. Thus $e$ is an upper edge of $b$ and a lower edge of $a$.

Let $L^+$ be the line of $\cal A$ with the largest positive slope, and $L^-$ the line with the steepest negative slope. (From the assumptions, $L^+$ and $L^-$ are unique and distinct.) Then there is a unique unbounded cell $A$ of $\cal A$ that is bounded by $L^+$ and $L^-$ and has no cell adjacently above it, and similarly there is a unique unbounded cell $B$, also bounded by $L^+$ and $L^-$, which has no cell adjacently below. So $A$ is the unique source of $G$, and $B$ the unique sink.

Now I assign numbers/weights to the nodes of $G$ as follows. $A$ is assigned $1$. For any node $x$ of $G$, $x$ is assigned the sum of the weights of the nodes adjacently above $x$, i.e., with incoming arcs to $x$. Every node but $A$ has at least one incoming arc, and so this is well-defined.

Two $4$-line arrangements are shown below.

^{ $L^+$ and $L^-$ are colored green. }

Every node but $B$ has at least one outgoing arc, so the weight of a node is always propagated downward. Thus the sink node $B$ receives the highest weight. In (a), $B$ has weight $8$, whereas in $B$ it has weight $9$.

A basic question is:

. What is the largest possible weight assigned to the sink node $B$, over all arrangements of $n$ lines?Q1

Here is a more complicated arrangement of $8$ lines, whose sink node has weight $111$.

(This was calculated by hand, so apologies if there are errors. Thanks to Will Brian for catching one error.)

The more general question is:

. Has this graph and node-weighting scheme been studied before, perhaps in another context?Q2