A note on \(Oct_{1}^{+}\)-free graphs and \(Oct_{2}^{+}\)-free graphs

Let \(Oct_{1}^{+}\) and \(Oct_{2}^{+}\) be the planar and non-planar graphs that obtained from the Octahedron by 3-splitting a vertex respectively. For \(Oct_{1}^{+}\), we prove that a 4-connected graph is \(Oct_{1}^{+}\)-free if and only if it is \(C_{6}^{2}\), \(C_{2k+1}^{2}\) \((k \geq 2)\) or it is obtained from \(C_{5}^{2}\) by repeatedly 4-splitting vertices. We also show that a planar graph is \(Oct_{1}^{+}\)-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from \(\{K_{1}, K_{2} ,K_{3}\} \cup \mathscr{K} \cup \{Oct,L_{5} \}\), where \(\mathscr{K}\) is the set of graphs obtained by repeatedly taking the special 3-sums of \(K_{4}\). For \(Oct_{2}^{+}\), we prove that a 4-connected graph is \(Oct_{2}^{+}\)-free if and only if it is planar, \(C_{2k+1}^{2}\) \((k \geq 2)\), \(L(K_{3,3})\) or it is obtained from \(C_{5}^{2}\) by repeatedly 4-splitting vertices.