Before revision:  We also prove a relaxed version of the conjecture that the size of the separators should not be larger than 2 \cdot tw +  1 when tw is the treewidth\le  .
After revision:  We also prove a relaxed version of the conjecture that the size of the separators should not be larger than 2 \cdot tw + 1 when tw is the treewidth\le 2tw(G), where tw(G) is the treewidth of G. (2) For any c < 2, there exists a graph G such that any optimal top separator S of G have |S| > c \cdot tw(G), i.e., the first result gives a tight bound on the size of an optimal top separator .