$K_{4}$-Minor-Free Induced Subgraphs of Sparse Connected Graphs

TL;DR

This paper proves that in any connected graph with m edges, there exists a small vertex set whose removal results in a graph with no K4 minor, establishing a tight bound related to the graph's treewidth.

Contribution

It provides a tight bound on the size of a vertex set needed to eliminate K4 minors in connected graphs, highlighting the importance of connectivity.

Findings

Bound of (3/16)(m+1) vertices for connected graphs

Bound of (1/5)m vertices for disconnected graphs

The bound is proven to be optimal

Abstract

We prove that every connected graph $G$ with $m$ edges contains a set $X$ of at most $\frac{3}{16}(m + 1)$ vertices such that $G-X$ has no $K_4$ minor, or equivalently, has treewidth at most $2$. This bound is best possible. Connectivity is essential: If $G$ is not connected then only a bound of $\frac{1}{5}m$ can be guaranteed.