Area lower bounds for truly integral Fary embeddings of trees and cacti

Determine area lower bounds for truly integral Fary embeddings of binary trees, specifically ascertain whether every truly integral Fary embedding of an n-vertex binary tree requires Ω(n^2) area, and derive area lower bounds for truly integral Fary embeddings of general trees and cactus graphs.

Background

The paper provides polynomial-time algorithms that construct truly integral Fary embeddings for stars, trees, and cacti, together with explicit polynomial upper bounds on grid area. For binary trees, the authors obtain quadratic-area embeddings; for general trees, bounds depend on the number of leaves and depth; and for cacti, the bounds are polynomial with additional blow-up in the presence of many triangles.

While upper bounds are established, lower bounds are not. The authors pose explicit questions about minimal area requirements, conjecturing that binary trees may require Ω(n^2) area and asking for lower bounds for more general trees and cacti.