NERVES, MINORS, AND PIERCING NUMBERS

Abstract

We make the first step towards a "nerve theorem" for graphs. Let G be a simple graph and let F be a family of induced subgraphs of G such that the intersection of any members of F is either empty or connected. We show that if the nerve complex of 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 F be a finite family of open connected sets in the plane such that the intersection of any members of F is either empty or connected. If among any p >= 3 members of F there are some three that intersect, then there is a set of C points which intersects every member of F, where C is a constant depending only on p.