Generate connected induced subgraphs as the satisfying assignments to a SAT instance I want a SAT instance (in CNF) whose set of satisfying assignments are the connected induced subgraphs of a given input graph. A general solution would be helpful, but I really only need this when the input is a subgraph of a 2D grid graph.

Of course, I want the size of the SAT instance to be polynomial in the size of the input graph, so that rules out ideas like encoding every path between each pair of vertices.

Edit: The answer I needed and accepted allows the use of extra variables to formulate a SAT instance.  Interestingly, @DavidEppstein shows that if no new variables are allowed, the required SAT instance cannot, in general, be found.
It's not possible if the variables are restricted to be only the given ones for the graph vertices. Consider a graph $G$ that has $2n+2$ vertices: $x$, $u_i$, $v_i$, and $y$ for $1\le i\le n$, formed by the union of $n$ edge-disjoint paths $x$--$u_i$--$v_i$--$y$. Also consider only the subgraphs that include both $x$ and $y$: if the whole graph has a small CNF formula for connected subgraphs, then so does this subset of subgraphs, obtained by setting the variables for $x$ and $y$ to true and then simplifying the formula.

The connected subgraphs that include both $x$ and $y$ do have a very simple $2$-DNF formula:
$$\bigvee\limits_{i=1}^n(u_i\wedge v_i).$$
However, when you convert this to its (unique) CNF form, it has $2^n$ terms (one for each way of excluding either $u_i$ or $v_i$ from the subgraph, for each $i$). So no polynomial-size CNF formula exists.

$G$ is not a grid graph but the same construction can easily be embedded into a grid by expanding $x$ and $y$ into larger connected subsets of grid points. As above, the whole graph can't have a small CNF formula, because if it did then you'd also have a small CNF formula for the connected subgraphs that include all the grid points representing $x$ and $y$, but we can write down the formula for this subset of the subgraphs and it does not have polynomial size.