Planar graphs in blowups of fans

TL;DR

This work shows that every $n$-vertex planar graph can be embedded in an $O(\sqrt{n}\log^2 n)$-blowup of a fan, which is equivalent to removing a vertex set $X$ of size $O(\sqrt{n}\log^2 n)$ so that the remainder has bandwidth $O(\sqrt{n}\log^2 n)$. The authors extend this to any proper minor-closed class and develop a bandwidth-flexibility framework using graph products, local sparsification, and volume-preserving Euclidean contractions, with a key distance function $d^*$ balancing contraction and density. Three core ingredients drive the results: a local sparsification lemma to bound local density after removing a small set, a generalized embedding-contraction method linking density to bandwidth, and a robust use of strong graph products to transfer these bounds across minor-closed classes. The work also provides specialized bounds for graphs on surfaces, $k$-planar graphs, and $(g,k)$-planar graphs, highlighting a unified approach to large-scale graph containment in fan-blowups and offering tools for algorithmic applications relying on low-bandwidth structure. Overall, the paper advances a cohesive framework connecting blowup structure, bandwidth, and minor-closed classes through a blend of decompositions, embeddings, and product-graph techniques.

Abstract

We show that every $n$-vertex planar graph is contained in the graph obtained from a fan by blowing up each vertex by a complete graph of order $O(\sqrt{n}\log^2 n)$. Equivalently, every $n$-vertex planar graph $G$ has a set $X$ of $O(\sqrt{n}\log^2 n)$ vertices such that $G-X$ has bandwidth $O(\sqrt{n}\log^2 n)$. We in fact prove the same result for any proper minor-closed class, and we prove more general results that explore the trade-off between $X$ and the bandwidth of $G-X$. The proofs use three key ingredients. The first is a new local sparsification lemma, which shows that every $n$-vertex planar graph $G$ has a set of $O((n\log n)/δ)$ vertices whose removal results in a graph with local density at most $δ$. The second is a generalization of a method of Feige and Rao that relates bandwidth and local density using volume-preserving Euclidean embeddings. The third ingredient is graph products, which are a key tool in the extension to any proper minor-closed class.

Figures (6)

• Figure 1: The $5$-blowup of a $6$-vertex fan.
• Figure 2: The strong product of a tree $H$ and a path $P$.
• Figure 3: Obstacles not in $X_{i,j}$ can interact to create excessively large distances between vertices in $Q_{i,j}$.
• Figure 4: The result of decomposing the graph $H\boxtimes P$ in \ref{['strong_product_fig']} with $\Delta=4$, $r_H=2$, and $r_P=3$.
• Figure 5: A diagram of how $G$ is turned into $G'$. The left graph is $G$, the right graph is $G'$. On the left, the vertices in the red section are $B_r$, where $r$ is the centre. The vertices in the blue section (all vertices) are $B_s$. Note that, on the right, each vertex in $B_s\setminus B_r=B_s'$ has been replaced by three new vertices, each adjacent to a different vertex of $B_r=\{a,b,c\}$, and that $B_r$ has become a clique. In this case, $M_v=\{v_a,v_b,v_c\}$ (and similar for $M_w,M_x,M_y,M_z$).

Theorems & Definitions (83)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Remark 6
• Lemma 7
• proof
• Theorem 8