Planar Straight-line Realizations of 2-Trees with Prescribed Edge Lengths

We study a classic problem introduced thirty years ago by Eades and Wormald. Let $G=(V,E,\lambda)$ be a weighted planar graph, where $\lambda: E \rightarrow \mathbb{R}^+$ is a length function. The Fixed Edge-Length Planar Realization problem (FEPR for short) asks whether there exists a planar straight-line realization of $G$, i.e., a planar straight-line drawing of $G$ where the Euclidean length of each edge $e \in E$ is $\lambda(e)$. Cabello, Demaine, and Rote showed that the FEPR problem is NP-hard, even when $\lambda$ assigns the same value to all the edges and the graph is triconnected. Since the existence of large triconnected minors is crucial to the known NP-hardness proofs, in this paper we investigate the computational complexity of the FEPR problem for weighted $2$-trees, which are $K_4$-minor free. We show its NP-hardness, even when $\lambda$ assigns to the edges only up to four distinct lengths. Conversely, we show that the FEPR problem is linear-time solvable when $\lambda$ assigns to the edges up to two distinct lengths, or when the input has a prescribed embedding. Furthermore, we consider the FEPR problem for weighted maximal outerplanar graphs and prove it to be linear-time solvable if their dual tree is a path, and cubic-time solvable if their dual tree is a caterpillar. Finally, we prove that the FEPR problem for weighted $2$-trees is slice-wise polynomial in the length of the longest path.