[Paper Review] Graphs without large $K_{2,n}$-minors

Lochet, William, Sharma, Roohani|arXiv (Cornell University)|Feb 5, 2017

Advanced Graph Theory Research2 references3 citations

TL;DR

This paper characterizes graphs without large $K_{2,n}$-minors by showing they are built via bounded 2-sums of two graph types: type-I graphs (a generalization of outerplanar graphs with limited chord crossings) and augmentations of bounded-size graphs. The key result is that every sufficiently large 3-connected graph with minimum degree ≥6, every 4-connected graph with a high-degree vertex, and every sufficiently large 5-connected graph must contain a $K_{2,n}$-minor for large $n$, establishing structural thresholds for minor containment.

ABSTRACT

In the F-Deletion problem, where F is a fixed finite family of graphs, the input is a graph G and an integer k, and the goal is to determine if there exists a set of at most k vertices whose deletion results in a graph that does not contain any graph of F as a minor. The F-Deletion problem encapsulates a large class of natural and interesting graph problems like Vertex Cover, Feedback Vertex Set, Treewidth-η Deletion, Treedepth-η Deletion, Pathwidth-η Deletion, Outerplanar Deletion, Vertex Planarization and many more. We study the F-Deletion problem from the kernelization perspective. In a seminal work, Fomin et al. [FOCS 2012] gave a polynomial kernel for this problem when the family F contains at least one planar graph. The asymptotic growth of the size of the kernel is not uniform with respect to the family F: that is, the size of the kernel is k^{f(F)}, for some function f that depends only on F. Later Giannopoulou et al. [TALG 2017] showed that the non-uniformity in the kernel size bound is unavoidable as Treewidth-η Deletion cannot admit a kernel of size 𝒪(k^{(η+1)/2 - ε}), for any ε > 0, unless NP ⊆ coNP/poly. On the other hand it was also shown that Treedepth-η Deletion admits a uniform kernel of size f(F) ⋅ k⁶ depicting that there are subclasses of F where the asymptotic kernel sizes do not grow as a function of the family F. This work led to the question of determining classes of F where the problem admits uniform polynomial kernels. In this paper, we show that if all the graphs in F are connected and ℱ contains K_{2,p} (a bipartite graph with 2 vertices on one side and p vertices on the other), then the problem admits a uniform kernel of size f(F) ⋅ k^10. The graph K_{2,p} is one natural extension of the graph θ_p, where θ_p is a graph on two vertices and p parallel edges. The case when F contains θ_p has been studied earlier and serves as (the only) other example where the problem admits a uniform polynomial kernel.

Motivation & Objective

• To characterize the structure of graphs that do not contain a large $K_{2,n}$-minor for any given $n$.
• To extend the known structural characterization of $K_{1,n}$-minor-free graphs to the more complex $K_{2,n}$-minor case.
• To establish sufficient connectivity conditions under which $K_{2,n}$-minors must exist in large graphs.
• To formalize a decomposition theorem using 2-sums and bounded augmentations to describe $K_{2,n}$-minor-free graphs.

Proposed method

• Introduces type-I graphs as a generalization of outerplanar graphs, defined by restricted chord crossings on a Hamiltonian cycle.
• Defines two types of augmentations: fans and strips, which are added to bounded-size graphs to form the second class of building blocks.
• Uses 2-sum operations to combine graphs, preserving structural constraints while building complex $K_{2,n}$-minor-free graphs.
• Applies induction and minor contraction/deletion arguments to show that type-I graphs avoid $K_{2,5}$-minors.
• Employs a recursive minor-avoidance argument in 2-sums, proving that if component graphs avoid $K_{2,m}$ and $K_{2,n}$-minors, then the 2-sum avoids $K_{2,mn}$-minors.
• Uses edge-separating sets in strips and fans to bound the number of disjoint connections, enabling exponential upper bounds on minor size.

Experimental results

Research questions

• RQ1 What structural properties characterize graphs that do not contain a large $K_{2,n}$-minor for a given $n$?
• RQ2 Can the class of $K_{2,n}$-minor-free graphs be decomposed using a finite set of building blocks and operations?
• RQ3 What connectivity thresholds force the presence of $K_{2,n}$-minors in large graphs?
• RQ4 How does the 2-sum operation affect the minor-closure properties of graphs with bounded-size augmentations?

Key findings

• Every $K_{2,n}$-minor-free graph can be constructed via a bounded number of 2-sum operations from type-I graphs and augmentations of bounded-size graphs.
• Type-I graphs do not contain a $K_{2,5}$-minor, establishing a structural barrier for minor containment.
• The 2-sum of two $K_{2,m}$- and $K_{2,n}$-minor-free graphs is $K_{2,mn}$-minor-free, enabling recursive minor-avoidance bounds.
• Every augmentation of a graph on at most $n$ vertices avoids a $K_{2,f_{6.3}(n)}$-minor, where $f_{6.3}(n) = 10n \cdot 2^n$, providing an exponential upper bound.
• For any fixed $n$, every sufficiently large 3-connected graph with minimum degree at least six contains a $K_{2,n}$-minor.
• Every 4-connected graph with a sufficiently high-degree vertex and every sufficiently large 5-connected graph must contain a $K_{2,n}$-minor for large $n$.