The extremal functions for triangle-free graphs with excluded minors

We prove two results:

1. A graph $G$ on at least seven vertices with a vertex $v$ such that $G-v$ is planar and $t$ triangles satisfies $\mathrm{\mid }E(G)\mathrm{\mid }\le 3\mathrm{\mid }V(G)\mathrm{\mid }-9+t\mathrm{/}3$.
2. For $p=2,3,\dots ,9$, a triangle-free graph $G$ on at least $2p-5$ vertices with no $K_p$-minor satisfies $\mathrm{\mid }E(G)\mathrm{\mid }\le (p-2)\mathrm{\mid }V(G)\mathrm{\mid }-(p-2{)}^2$.