Given a nonplanar graph $G$ drawn in the plane with crossings. Does there exist a small subset $S$ of edges of $G$ such that after the removal of all edges that intersect or share an endpoint of an edge of $S$, each component of the remaining graph has at most $\frac{2}{3}|V(G)|$ vertices.

Given a nonplanar graph $G$ drawn in the plane with crossings. Does there exist a small ($o(|V(G)|$) subset $S$ of edges of $G$ such that after the removal of all edges that intersect or share an endpoint of an edge of $S$, each component of the remaining graph has at most $\frac{2}{3}|V(G)|$ vertices.

Given a nonplanar graph $G$ drawn in the plane with crossings. Does there exist a small subset $S$ of edges of $G$ such that after the removal of all edges that intersect or share an endpoint of an edge of $S$, each component of the remaining graph

Given a nonplanar graph $G$ drawn in the plane with crossings. Does there exist a small subset $S$ of edges of $G$ such that after the removal of all edges that intersect or share an endpoint of an edge of $S$, each component of the remaining graph has at most $\frac{2}{3}|V(G)|$ vertices.