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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 K 4 -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. [ABSTRACT FROM AUTHOR]

A simple topological graph is k -quasiplanar (k ≥ 2) if it contains no k pairwise crossing edges, and k -planar if no edge is crossed more than k times. In this paper, we explore the relationship between k -planarity and k -quasiplanarity to show that, for k ≥ 2 , every k -planar simple topological graph can be transformed into a (k + 1) -quasiplanar simple topological graph. [ABSTRACT FROM AUTHOR]