Nerves, minors, and piercing numbers

We make the first step towards a "nerve theorem" for graphs. Let $G$ be a simple graph and let $\mathcal{F}$ be a family of induced subgraphs of $G$ such that the intersection of any members of $\mathcal{F}$ is either empty or connected. We show that if the nerve complex of $\mathcal{F}$ has non-vanishing homology in dimension three, then $G$ contains the complete graph on five vertices as a minor. As a consequence we confirm a conjecture of Goaoc concerning an extension of the planar $(p,q)$ theorem due to Alon and Kleitman: Let $\mathcal{F}$ be a finite family of open connected sets in the plane such that the intersection of any members of $\mathcal{F}$ is either empty or connected. If among any $p \geq 3$ members of $\mathcal{F}$ there are some three that intersect, then there is a set of $C$ points which intersects every member of $\mathcal{F}$, where $C$ is a constant depending only on $p$.