I was not aware of TRVGs but I was immediately reminded of [interval graphs](https://en.wikipedia.org/wiki/Interval_graph) and the related [interval trees](https://en.wikipedia.org/wiki/Interval_tree) of which TRVGs seem to be a generalisation of in two dimensions. Especially with this constraint on the rectangles to be aligned to the axes. Each rectangle then could represent two coupled intervals and the condition on the existence of the edge is translated to whether or not they overlap in a given dimension.
That’s a great point, and your idea immediately suggests other generalizations. In particular, one could have an arrangement of cuboids in three dimensions with analogous rules. Given a finite graph G, I wonder what conditions (if any) on G are necessary for there to exist an integer k such that G is a k-cuboid visibility graph? The minimal such k, if one exists, also gives a notion of dimension for G.
I'd never heard of RVGs, let alone TRVGs. What a cool idea, thanks for sharing!
Thanks! I was a bit worried this post might be too unrelatable, so I’m glad you liked it.
Yeah, made me grin and think of getting some graph paper and trying it with some tree structures. Or maybe writing some Python to automate it… 🤔
I was not aware of TRVGs but I was immediately reminded of [interval graphs](https://en.wikipedia.org/wiki/Interval_graph) and the related [interval trees](https://en.wikipedia.org/wiki/Interval_tree) of which TRVGs seem to be a generalisation of in two dimensions. Especially with this constraint on the rectangles to be aligned to the axes. Each rectangle then could represent two coupled intervals and the condition on the existence of the edge is translated to whether or not they overlap in a given dimension.
That’s a great point, and your idea immediately suggests other generalizations. In particular, one could have an arrangement of cuboids in three dimensions with analogous rules. Given a finite graph G, I wonder what conditions (if any) on G are necessary for there to exist an integer k such that G is a k-cuboid visibility graph? The minimal such k, if one exists, also gives a notion of dimension for G.