# Drawing Graphs as Spanners

We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph G, the goal is to construct a straight-line drawing Γ of G in the plane such that, for any two vertices u and v of G, the ratio between the minimum length of any path from u to v and the Euclidean distance between u and v is small. The maximum such ratio, over all pairs of vertices of G, is the spanning ratio of Γ. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio 1, a proper straight-line drawing with spanning ratio 1, and a planar straight-line drawing with spanning ratio 1 are NP-complete, ∃ℝ-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio 1 to spanning ratio 1+ϵ allows us to draw every graph. Namely, we prove that, for every ϵ>0, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than 1+ϵ. Third, our drawings with spanning ratio smaller than 1+ϵ have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.

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08/31/2017

### On the Edge-length Ratio of Outerplanar Graphs

We show that any outerplanar graph admits a planar straightline drawing ...
research
07/20/2023

### Manipulating Weights to Improve Stress-Graph Drawings of 3-Connected Planar Graphs

We study methods to manipulate weights in stress-graph embeddings to imp...
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08/09/2019

### On the Edge-Length Ratio of Planar Graphs

The edge-length ratio of a straight-line drawing of a graph is the ratio...
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09/24/2019

### On the edge-length ratio of 2-trees

We study planar straight-line drawings of graphs that minimize the ratio...
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10/23/2019

### Simplified Emanation Graphs: A Sparse Plane Spanner with Steiner Points

Emanation graphs of grade k, introduced by Hamedmohseni, Rahmati, and Mo...
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08/20/2020

### Plane Spanning Trees in Edge-Colored Simple Drawings of K_n

Károlyi, Pach, and Tóth proved that every 2-edge-colored straight-line d...
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05/05/2020

### Grid Drawings of Graphs with Constant Edge-Vertex Resolution

We study the algorithmic problem of computing drawings of graphs in whic...