What is the optimal polygonal complexity for 3D floorplans for chordal graphs?

Question 8.7 remains interesting even for interval graphs.

In addition to polygonal complexity, the volume of the drawing, which can be mea-sured by the size of the underlying grid or the number of grid planes, is another important quality measure.

In Corollary 7.2, we showed that interval graphs admit a 2-layer 3D floorplan which fits into a O(1)× O(k) × O(|V (G)|) grid, where k is the size of the maximum clique in G. In other words, it requires O(|V (G)|) grid planes. We conjecture that it is optimal.

Question 8.8. Is it possible to construct a 3D floorplan using o(|V (G)|) grid planes for every interval graph G?

The following research directions are also interesting:

1. improving the constant behind Corollary 7.2; and

2. reducing the number of grid planes in our algorithm for constructing 3D floorplans of chordal graphs.

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