Rooted $K_4$-Minors

TL;DR

This paper characterizes when a graph contains a rooted $K_4$-minor at four specified vertices, identifying six classes of obstructions and providing conditions for 4-connected non-planar and 3-connected planar graphs.

Contribution

It provides a precise characterization of rooted $K_4$-minors through six classes of obstructions, extending understanding of graph minors with specific root vertices.

Findings

Six classes of obstructions identified

4-connected non-planar graphs always contain the rooted $K_4$-minor

3-connected planar graphs contain the rooted $K_4$-minor iff roots are not on a single face

Abstract

Let $a,b,c,d$ be four vertices in a graph $G$. A \emph{$K_4$-minor rooted} at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connected planar graph contains a $K_4$-minor rooted at $a,b,c,d$ if and only if $a,b,c,d$ are not on a single face.