Search

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]

In this paper we study planar morphs between planar straight-line grid drawings of trees. A morph consists of a sequence of morphing steps, where in a morphing step vertices move along straight-line trajectories at constant speed. We show how to construct planar morphs that simultaneously achieve a reduced number of morphing steps and a polynomially-bounded resolution. We assume that both the initial and final drawings lie on the grid and we ensure that each morphing step produces a grid drawing; further, we consider both upward drawings of rooted trees and drawings of arbitrary trees. [ABSTRACT FROM AUTHOR]