On the Edge-Length Ratio of Planar Graphs

Abstract

The edge-length ratio of a straight-line drawing of a graph is the ratio between the lengths of the longest and of the shortest edge in the drawing. The planar edge-length ratio of a planar graph is the minimum edge-length ratio of any planar straight-line drawing of the graph.In this paper, we study the planar edge-length ratio of planar graphs. We prove that there exist n-vertex planar graphs whose planar edge-length ratio is in Ω(n); this bound is tight. We also prove upper bounds on the planar edge-length ratio of several families of planar graphs, including series-parallel graphs and bipartite planar graphs.

On the edge-length ratio of 2-trees

“…Recently, Borrazzo and Frati showed (among other results) that any 2-tree on n vertices could be drawn with edge-length ratio at most n 0.695 [1]. As our construction provides a logarithmic lower bound, this naturally rises the question what is the right asymptotic growth of the edge-length ratio of 2-trees.…”

“…For given r we argue first that a sufficiently large 2-tree drawn with edges of length at most r contains a triangle with area at most 1 2 . Then, inside this triangle of small area we build a sequence of triangles with perimeters decreasing by 1 2 in each step, which results of a triangle with edge of length less than 1. We consider a special subclass G = {G 0 , G 1 , .…”

“…. , k} the triangle ∆ i is a separating triangle of level i, and for each i > 1, in addition, it holds that A(∆ i ) ≤ 1 2 A(∆ i−1 ).…”

“…We call thin any triangle with edges of length at least 1 and area at most 1 4 . Any thin triangle has height at most 1 2 and hence one obtuse angle of size at least 2π 3 and two acute angles, each of size at most π 6 . Lemma 3.…”

Planar Straight-line Realizations of 2-Trees with Prescribed Edge Lengths

“…There is a vast amount of research on 2-trees in Graph Drawing (e.g., in [18,25,28,33,39]). The edge lengths of 2-trees have been studied in [9,10].…”