Planar Straight-line Realizations of 2-Trees with Prescribed Edge Lengths
Carlos Alegr\'ia, Manuel Borrazzo, Giordano Da Lozzo, Giuseppe Di, Battista, Fabrizio Frati, Maurizio Patrignani

TL;DR
This paper investigates the computational complexity of the Fixed Edge-Length Planar Realization problem for specific classes of graphs, showing NP-hardness for weighted 2-trees but efficient solutions under certain conditions.
Contribution
It proves NP-hardness of FEPR for weighted 2-trees with up to four edge lengths and provides linear-time algorithms for cases with limited edge lengths or specific graph structures.
Findings
NP-hardness for weighted 2-trees with up to four edge lengths
Linear-time algorithms for FEPR with up to two edge lengths
Efficient solutions for weighted maximal outerplanar graphs with specific dual tree structures
Abstract
We study a classic problem introduced thirty years ago by Eades and Wormald. Let be a weighted planar graph, where is a length function. The Fixed Edge-Length Planar Realization problem (FEPR for short) asks whether there exists a planar straight-line realization of , i.e., a planar straight-line drawing of where the Euclidean length of each edge is . Cabello, Demaine, and Rote showed that the FEPR problem is NP-hard, even when 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 -trees, which are -minor free. We show its NP-hardness, even when assigns to the edges only up to…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research
