A Note on Oct1+-Minor-Free Graphs and Oct2+-Minor-Free Graphs

Let [Formula: see text] and [Formula: see text] be the planar and non-planar graphs, respectively, obtained from the Octahedron by 3-splitting a vertex. For [Formula: see text], we prove that if a 4-connected graph is [Formula: see text]-minor-free, then it is [Formula: see text], [Formula: see text] [Formula: see text] or it is obtained from [Formula: see text] by repeatedly 4-splitting vertices. We also show that a planar graph is [Formula: see text]-minor-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from [Formula: see text], where [Formula: see text] is the set of graphs obtained by repeatedly taking the special 3-sums of [Formula: see text] and [Formula: see text] is the graph obtained from two 5-cycles [Formula: see text], [Formula: see text] by adding the five edges [Formula: see text] for all [Formula: see text]. For [Formula: see text], we prove that if a 4-connected graph is [Formula: see text]-minor-free, then it is planar, [Formula: see text] [Formula: see text], the line graph of [Formula: see text] or it is obtained from [Formula: see text] by repeatedly 4-splitting vertices.