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 , lambda) be a weighted planar graph, where lambda : 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 straightline drawing of G where the Euclidean length of each edge e is an element of 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 the NP-hardness of the problem, 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 large path. (c) 2023 Published by Elsevier Ltd.