The $K_{n+5}$ and $K_{3^2,1^n}$ families are obstructions to $n$-apex
  We give evidence in support of a conjecture that the $\nabla$Y$\nabla$ families of $K_{n+5}$ and $K_{3^2,1^n}$ are obstructions for the $n$-apex property. 
Thomas W. Mattman and Mike Pierce
math.CO