# connected subgraphs having minimum expansion

If I have a connected graph $$G$$, is it true that subgraphs(corresponding to cuts) having minimum conductance or expansion(or other connectivity measures) should be connected? In other words if $$S$$ is a subgraph for which $$\phi(G) = |\delta(S)|/|S|$$ does this imply that $$S$$ and $$V-S$$ are connected?

NOTE: Assuming $$S$$ having at least two connected components (and $$S_1$$ one of them) I tried to use the connectedness of the graph to make cut $$S' = S - S_1$$ reduce the nominator of the fraction and controlling min(|S'|,|V_S'|) but it seems that this does't work!

• The answer is no. For example consider the star graphs. – Mahdi Jan 6 at 8:59