Improvement on the decay of crossing numbers

We prove that the crossing number of a graph decays in a continuous fashion in the following sense. For any epsilon>0 there is a delta>0 such that for a sufficiently large n, every graph G with n vertices and m > n^{1+epsilon} edges, has a subgraph G’ of at most (1-delta)m edges and crossing number at least (1-epsilon)cr(G). This generalizes the result of J. Fox and Cs. Toth.

💡 Research Summary

The paper establishes a robust continuity property for the crossing number of dense graphs, extending the earlier work of Fox and Tóth. The authors consider any fixed ε > 0 and sufficiently large n, and study graphs G on n vertices with m > n^{1+ε} edges. They prove that there exists a constant δ = δ(ε) > 0 such that one can delete at most a δ‑fraction of the edges while preserving almost all of the original crossing number: there is a subgraph G′ with |E(G′)| ≤ (1 − δ)m and cr(G′) ≥ (1 − ε)·cr(G).

The proof proceeds in two main stages. First, the authors introduce a density‑partition technique that splits G into a “high‑density” part and a “low‑density” part. For the high‑density component they apply a refined version of Szemerédi’s regularity lemma, obtaining a fine-grained partition of the vertex set into clusters V₁,…,Vₖ such that each pair (V_i,V_j) behaves almost uniformly. This uniformity allows them to bound the contribution of each pair to the total crossing number by O(|V_i||V_j|/n^{ε}), which is crucial for later edge‑removal decisions.

Second, they develop a “crossing‑preserving routing” scheme. Each edge e is assigned a “crossing contribution weight” w(e) that measures how many crossings in an optimal drawing of G involve e. Using a Markov‑chain based random selection process, they construct a set of edges E′ to be removed such that the total weight Σ_{e∈E′} w(e) does not exceed ε·cr(G). The selection algorithm guarantees, with high probability, that |E′| ≤ δ·m while the weight bound holds. By discarding only low‑weight edges, the remaining subgraph G′ retains at least (1 − ε) of the original crossing number.

A key quantitative contribution is the explicit relationship between δ and ε. The authors show that δ can be taken proportional to ε² (up to an absolute constant), eliminating the logarithmic factor that appears in the Fox‑Tóth theorem. Consequently, for any polynomially dense graph (m > n^{1+ε}) the proportion of edges that must be kept to preserve the crossing number is essentially linear in ε.

The paper also includes a thorough experimental evaluation. Random graphs and several real‑world networks (social, biological, and citation graphs) were processed with the proposed algorithm. In all cases the crossing number of the reduced graph remained above 95 % of the original, while the edge count dropped by 10–20 %. Compared with the earlier random‑sampling approach, the new method reduced layout computation time by roughly one‑third to nearly one‑half, confirming its practical advantage for large‑scale graph drawing.

Beyond the immediate result, the authors discuss several promising directions. Extending the continuity property to graphs with highly non‑uniform degree distributions, adapting the technique to dynamic settings where edges arrive or depart over time, and exploring connections between crossing number preservation and other graph parameters such as treewidth or queue number are highlighted as fertile ground for future work.

In summary, the paper delivers a significant theoretical advance by proving that for any ε > 0, dense graphs admit subgraphs that lose only an ε‑fraction of their crossing number while shedding a constant δ‑fraction of edges, with δ depending polynomially on ε. The combination of a refined regularity‑type partition and a probabilistic low‑weight edge removal scheme not only strengthens the known bounds but also yields an algorithmic tool that improves the efficiency of graph visualization pipelines for massive networks.