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 $\Gamma$ 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 $\Gamma$. 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, $\exists \mathbb R$-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 $1+\epsilon$ allows us to draw every graph. Namely, we prove that, for every $\epsilon>0$, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than $1+\epsilon$. Third, our drawings with spanning ratio smaller than $1+\epsilon$ 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.

💡 Research Summary

The paper “Drawing Graphs as Spanners” investigates how to embed abstract graphs in the plane as straight‑line geometric spanners, i.e., drawings in which the length of the shortest path between any two vertices is close to their Euclidean distance. The authors focus on three increasingly relaxed notions of a drawing with spanning ratio 1 (the optimal value) and then on drawings with ratio 1 + ε for any ε > 0.

First, they show that deciding whether a graph admits a straight‑line drawing with spanning ratio 1 is NP‑complete. The key observation (Lemma 1) is that a spanning‑ratio‑1 drawing forces the existence of a Hamiltonian path, because vertices can be ordered by increasing x‑coordinate and each consecutive pair must be adjacent. Since Hamiltonian‑path existence is NP‑complete, the drawing problem inherits this hardness (Theorem 1).

Next they consider “proper” drawings, where no two vertices coincide and no edge passes through another vertex. They prove (Lemma 2) that a graph has a proper spanning‑ratio‑1 drawing iff it is a point‑visibility graph of some point set. Recognizing point‑visibility graphs is known to be ∃ℝ‑complete, so the proper‑drawing decision problem is ∃ℝ‑complete (Theorem 2).

In contrast, requiring the drawing to be planar (edges may intersect only at common endpoints) makes the problem tractable. Dujmović et al. (2022) characterized graphs that admit a planar straight‑line drawing where every pair of vertices is joined by a straight segment; these are exactly the five graph classes shown in Figure 1. Since spanning‑ratio‑1 planar drawings are precisely those where every pair is connected by a straight segment, recognizing such graphs can be done in linear time (Theorem 3).

The authors then relax the requirement to spanning ratio 1 + ε for any fixed ε > 0. They prove that every connected planar graph admits a planar straight‑line drawing with ratio < 1 + ε (Theorem 4). The construction builds on a canonical ordering of a maximal planar supergraph G of the input H. By inserting vertices one by one while keeping the outer face a simple polygon, they can place each new vertex sufficiently close to the line segment joining its two outer‑face neighbors. This guarantees that any path that follows the inserted vertices stays within a (1 + ε) factor of the Euclidean distance. A similar argument works for non‑planar graphs, yielding proper (non‑planar) drawings with the same bound.

However, achieving a constant spanning ratio inevitably forces a large edge‑length ratio (the ratio of longest to shortest edge). The authors show that for stars, any constant‑ratio drawing must have exponential edge‑length ratio. More generally, they introduce the graph toughness τ(G) and prove that if τ(G) is small, then any constant‑ratio drawing must have edge‑length ratio at least exp(Θ(1/τ(G))) (Theorem 5). Conversely, families with bounded toughness admit proper drawings with polynomial edge‑length ratio and constant spanning ratio (Theorem 6). They also give a constructive polynomial‑ratio planar drawing for bounded‑degree trees.

In summary, the paper establishes a sharp dichotomy: exact spanning ratio 1 is computationally hard (NP‑complete or ∃ℝ‑complete) unless planarity is imposed, while allowing an arbitrarily small additive slack (1 + ε) makes every graph drawable as a geometric spanner. The trade‑off between spanning ratio and edge‑length ratio is governed by graph toughness, providing a nuanced understanding of when high‑quality spanner drawings are feasible. These results have implications for graph visualization, network layout, and the theory of geometric spanners.