The problem seems nice and difficult to me.

First of all, let’s see if I understood correctly your game.

At the beginning, weeds are everywhere along the roads. You pick a point in the green grid and start mowing weeds moving in one direction. Instantly, as you clear the road, weeds from behind you starts spreading along the road at speed 1. When you arrive at a crossroad, you must select just one possible way, but if spreading weeds arrive at a crossroad, they spread in all directions.

Assuming that all this is right, let’s see…

It seems clear that a large enough speed would do the job. A trivial case is the one where you reduced all weeds to within one “step” of the grid, with no crossroads involved. Then you can make it if the speed is more than $\frac L\ell$, where $L$ is the length of the weeds segment and $\ell$ is the distance between the weeds segment endpoint and the closest crossroad. But this is only a trivial sufficient condition, of course. Assuming that at the crossroad four paths intersect, you can still make it with $\ell$ very small if you can chop the three new branches arising one at a time. This easily leads to the (very lazy…) bound $v>7$, so the zero-order step is: if you confined weeds to a single step of the grid, you can do it if your speed is 7 or more.

I’ll try to come back later to think about more interesting cases.

The problem seems nice and difficult to me.

First of all, let’s see if I understood correctly your game.

At the beginning, weeds are everywhere along the roads. You pick a point in the green grid and start mowing weeds moving in one direction. Instantly, as you clear the road, weeds from behind you starts spreading along the road at speed 1. When you arrive at a crossroad, you must select just one possible way, but if spreading weeds arrive at a crossroad, they spread in all directions.

Assuming that all this is right, let’s see…

It seems clear that a large enough speed would do the job. A trivial case is the one where you reduced all weeds to within one “step” of the grid, with no crossroads involved. Then you can make it if the speed is more than $\frac L\ell$, where $L$ is the length of the weeds segment and $\ell$ is the distance between the weeds segment endpoint and the closest crossroad. But this is only a trivial sufficient condition, of course. Assuming that at the crossroad four paths intersect, you can still make it with $\ell$ very small if you can chop the three new branches arising one at a time. This easily leads to the (very lazy…) bound $v>7$, so the zero-order step is: if you confined weeds to a single step of the grid, you can do it if your speed is 7 or more.

I’ll try to come back later to think about more interesting cases.

Update: in the new version with only 2 squares, a better lazy bound can be given for the zero-step, that is $v>5$.

The problem seems nice and difficult to me.

First of all, let’s see if I understood correctly your game.

At the beginning, weeds are everywhere along the roads. You pick a point in the green grid and start mowing weeds moving in one direction. Instantly, as you clear the road, weeds from behind you starts spreading along the road at speed 1. When you arrive at a crossroad, you must select just one possible way, but if spreading weeds arrive at a crossroad, they spread in all directions.

Assuming that all this is right, let’s see…

It seems clear that a large enough speed would do the job. A trivial case is the one where you reduced all weeds to within one “step” of the grid, with no crossroads involved. Then you can make it if the speed is more than $\frac L\ell$, where $L$ is the length of the weeds segment and $\ell$ is the distance between the weeds segment endpoint and the closest crossroad. But this is only a trivial sufficient condition, of course. Assuming that at the crossroad four paths intersect, you can still make it with $\ell$ very small if you can chop the three new branches arising one at a time. This easily leads to the non-optimal bound $v>7$, so the zero-order step is: if you confined weeds to a single step of the grid, you can do it if your speed is 7 or more.

I’ll try to come back later to think about more interesting cases.

The problem seems nice and difficult to me.

First of all, let’s see if I understood correctly your game.

At the beginning, weeds are everywhere along the roads. You pick a point in the green grid and start mowing weeds moving in one direction. Instantly, as you clear the road, weeds from behind you starts spreading along the road at speed 1. When you arrive at a crossroad, you must select just one possible way, but if spreading weeds arrive at a crossroad, they spread in all directions.

Assuming that all this is right, let’s see…

It seems clear that a large enough speed would do the job. A trivial case is the one where you reduced all weeds to within one “step” of the grid, with no crossroads involved. Then you can make it if the speed is more than $\frac L\ell$, where $L$ is the length of the weeds segment and $\ell$ is the distance between the weeds segment endpoint and the closest crossroad. But this is only a trivial sufficient condition, of course. Assuming that at the crossroad four paths intersect, you can still make it with $\ell$ very small if you can chop the three new branches arising one at a time. This easily leads to the (very lazy…) bound $v>7$, so the zero-order step is: if you confined weeds to a single step of the grid, you can do it if your speed is 7 or more.

I’ll try to come back later to think about more interesting cases.