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In this paper we prove two main results about obstruction to graph planarity.One is that, if $G$ is a 3-connected graph with a $K_5$-minor and $T$ is atriangle of $G$, then $G$ has a $K_5$-minor $H$, such that $E(T)\cont E(H)$.  Other is that if $G$ is a 3-connected simple non-planar graph not isomorphicto $K_5$ and $e,f\in E(G)$, then $G$ has a minor $H$ such that $e,f\in E(H)$and, up to isomorphisms, $H$ is one of the four non-isomorphic simple graphsobtained from $K_{3,3}$ by the addiction of \,0, 1 or 2 edges. We generalizethis second result to the class of the regular matroids.

On $K_5$ and $K_{3,3}$-minors of graphs and regular matroids