A Characterization of $4$-Connected Graphs with no $ K_{3,3}+v $-Minor

Abstract

Let $ K_{3,3}+v $ be 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 characterize all $ 4 $-connected graphs that do not contain $ K_{3,3}+v $ as a minor.