  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.
