Preserving Terminal Distances using Minors

We introduce the following notion of compressing an undirected graph G with edge-lengths and terminal vertices $R\subseteq V(G)$. A distance-preserving minor is a minor G’ (of G) with possibly different edge-lengths, such that $R\subseteq V(G’)$ and the shortest-path distance between every pair of terminals is exactly the same in G and in G’. What is the smallest f*(k) such that every graph G with k=|R| terminals admits a distance-preserving minor G’ with at most f*(k) vertices? Simple analysis shows that $f*(k)\leq O(k^4)$. Our main result proves that $f*(k)\geq \Omega(k^2)$, significantly improving over the trivial $f*(k)\geq k$. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice.

💡 Research Summary

The paper investigates the fundamental limits of compressing an undirected weighted graph while exactly preserving the shortest‑path distances between a designated set of terminal vertices R. A “distance‑preserving minor” G′ of a graph G is defined as a minor (obtained by edge deletions, vertex deletions, and edge contractions) possibly with altered edge lengths, such that R⊆V(G′) and for every pair u,v∈R the distance in G equals the distance in G′. The central question is to determine the smallest function f⁎(k) (where k=|R|) such that every graph G admits a distance‑preserving minor with at most f⁎(k) vertices.

Upper bound. A straightforward construction takes, for each of the (\binom{k}{2}) terminal pairs, a shortest‑path subgraph and then takes the union of all these paths. In the worst case each path may contain O(k) distinct internal vertices, yielding an O(k⁴) bound on the number of vertices needed. This simple argument establishes that f⁎(k)≤O(k⁴).

Main lower bound. The authors dramatically improve the known trivial lower bound f⁎(k)≥k by proving f⁎(k)≥Ω(k²). The proof is based on a carefully crafted planar graph that forces a quadratic number of “essential” vertices. They place the k terminals on the boundary of a large grid and connect each pair of terminals by a shortest path that traverses a unique interior crossing point. Because each crossing point encodes the distance between a distinct pair of terminals, any minor that preserves all terminal distances must retain at least one vertex for each crossing. Consequently, any distance‑preserving minor must contain Ω(k²) vertices. This construction works even when the original graph is planar, showing that planarity does not help to reduce the size of a distance‑preserving minor.

Treewidth‑bounded graphs. In stark contrast, the paper shows that when the input graph has constant treewidth (for example, trees, series‑parallel graphs, or any graph that admits a tree decomposition of bounded width), a linear‑size distance‑preserving minor always exists. By exploiting the tree decomposition, the authors design a dynamic‑programming algorithm that selects a small set of representative vertices and contracts the rest while preserving all terminal distances. The resulting minor contains O(k) vertices, which matches the obvious lower bound and is optimal for this class.

Technical contributions.

1. Definition of distance‑preserving minors with flexible edge‑length reassignment.
2. A simple O(k⁴) universal upper bound obtained by union of all terminal‑pair shortest paths.
3. A planar “crossing‑grid” construction that forces Ω(k²) indispensable vertices, establishing the first super‑linear lower bound for general graphs.
4. An algorithmic framework for constant‑treewidth graphs that yields O(k)‑size minors, based on tree‑decomposition and careful contraction rules.

Implications and open problems. The results delineate a clear dichotomy: for unrestricted graphs the best possible compression is at least quadratic in the number of terminals, while for graphs of bounded treewidth the compression can be linear. This raises several natural questions: Is the true asymptotic behavior of f⁎(k) for general graphs Θ(k²) or does a higher lower bound exist? Can the O(k⁴) universal upper bound be improved, perhaps to O(k³) or even O(k² log k)? What are the precise bounds for other important graph families such as minor‑closed families, graphs of bounded genus, or graphs with bounded degree?

Overall, the paper makes a significant step toward understanding the limits of distance‑preserving graph compression, providing both a strong super‑linear lower bound and optimal linear constructions for a broad class of sparse graphs. The techniques introduced—particularly the crossing‑grid lower‑bound construction—are likely to be useful in related problems concerning graph minors, spanners, and metric embeddings.