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.