I'm going to prove the following statement:
$G$ is a $P_5$-free and $K_{1,3}$-free graph with $\vert G \vert \geq 7$, and $G \notin \mathcal{K}$, then $G$ is $2$-connected graph if and only if $G$ have two independent spanning trees, where $\mathcal{K}$ is a special class of graphs.
I believe this proposition should be correct, because I have proven most cases. But there remains one last case that has been troubling me for over a month:
Suppose $K$ is the largest dominating clique of $G$ and $\vert K \vert \geq 4$. $G \setminus K$ is a connect graph, and $N_K(G \setminus K) = V(K)$.
I attempted to discuss using the following method, but did not obtain any results:
Let $V(K) = \{ v_1, v_2, \dots, v_n \}$, $U_i = \{ u : u \in N_{G \setminus K}(v_i), N_K(u) = \{ v_i \} \}$, $U = \{ U_i \}$. Discuss based on the size of $U$.
Because this is the topic of my undergraduate thesis, I'm not sure if I can directly ask for the proof process online, but perhaps I can seek some ideas for the proof. Thank you very much!
