A note on $Oc{t}_1^{+}$-free graphs and $Oc{t}_2^{+}$-free graphs

Abstract

Let $Oc{t}_1^{+}$ and $Oc{t}_2^{+}$ be the planar and non-planar graphs that obtained from the Octahedron by 3-splitting a vertex respectively. For $Oc{t}_1^{+}$, we prove that a 4-connected graph is $Oc{t}_1^{+}$-free if and only if it is $C_6^2$, $C_{2k+1}^2$ $(k\ge 2)$ or it is obtained from $C_5^2$ by repeatedly 4-splitting vertices. We also show that a planar graph is $Oc{t}_1^{+}$-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from $\{K_1,K_2,K_3\}\cup \mathcal{K}\cup \{Oct,L_5\}$, where $\mathcal{K}$ is the set of graphs obtained by repeatedly taking the special 3-sums of $K_4$. For $Oc{t}_2^{+}$, we prove that a 4-connected graph is $Oc{t}_2^{+}$-free if and only if it is planar, $C_{2k+1}^2$ $(k\ge 2)$, $L(K_{3,3})$ or it is obtained from $C_5^2$ by repeatedly 4-splitting vertices.

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