Looking for a visual solution to connect some boxes in a grid where direction matters

I have 9 boxes, each with 1.5cm height and width. I want to place these boxes in a grid (for the sake of simpleness, let's put them in a 3x3 grid). The total size that my grid can hold up is around 7x7 cm.

Then, I would need to connect some of these boxes in the grid, in a way, where direction matters.

For example, if I get the following grid:

|1|2|3|
-------
|4|5|6|
-------
|7|8|9|

I would for example need to connect the box with id 6 to the box with id 3 and 5, representing that you can travel from 6 to 3 and 5 but not inversely(you can not travel from 3 to 6 and neither from 5 to 6).

My connections will always have to be one directional, so there can be no connection between 6 and 5 if there is already a connection from 5 to 6.

I hope that I didn't overcomplicate my question, and it is understandable:) I would prefer a solution which uses the white space between the boxes in the grid (I know that the alloted space is rather small), because there will be icons in each four corners of the boxes.

These boxes would be put on a card for a physical board game, so the end result should be visible when printed.

Anyone has any ideas on this?