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.
Categories: math.CO