  Among graphs with 13 edges, there are exactly three internally 4-connected
graphs which are $Oct^{+}$, cube+e and $ K_{3,3} +v$. A complete
characterization of all 4-connected graphs with no $Oct^{+}$-minor is given in
[John Maharry, An excluded minor theorem for the octahedron plus an edge,
Journal of Graph Theory 57(2) (2008) 124-130]. Let $K_{3,3}+v$ denote the graph
obtained by adding a new vertex $v$ to $K_{3,3}$ and joining $v$ to the four
vertices of a 4-cycle. In this paper, we determine all 4-connected graphs that
do not contain $K_{3,3}+v$ as a minor.
