Can planar graphs be chained?

Question

Let $G$ be a graph whose vertex set is described as the union of three disjoint sets $V = V_1 \cup V_2 \cup V_3$, and such that there are no edges between $V_1$ and $V_3$. Assuming the subgraphs induiced by $V_1 \cup V_2$ and $V_2 \cup V_3$ are planar, does this imply that $G$ is planar?

Answer 1

The complete bipartite graph $G=K_{3,3}$ is a non-planar graph. Let $v_1,v_3$ be two non-adjacent vertices of $G.$ Let $V_1=\{v_1\},\ V_3=\{v_3\},\ V_2=V(G)\setminus\{v_1,v_3\}.$ The graphs induced by $V_1\cup V_2$ and $V_2\cup V_3$ are isomorphic to $K_{2,3}$ which is planar.