Testing the planar straight-line realizability of 2-trees with prescribed edge lengths

We study a classic problem introduced thirty years ago by Eades and Wormald. Let G=(V,E,λ) be a weighted planar graph, where λ:E→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∈E is λ(e). Cabello, Demaine, and Rote showed that the FEPR problem is NP-hard, even when λ 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 K4-minor free. We show the NP-hardness of the problem, even when λ assigns to the edges only up to four distinct lengths. Conversely, we show that the FEPR problem is linear-time solvable when λ 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 large path.